GCSE Maths: Simultaneous equations: elimination (Higher)
Part of the GCSE Maths quadratic and simultaneous equations chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Simultaneous equations: elimination
One equation with two letters, such as x + y = 7, has lots of solutions: (1, 6), (2, 5), (3, 4), and so on. But if a second equation must be true at the same time, such as x − y = 1, there is usually only one pair of values that works for both: x = 4 and y = 3. Two equations that must both be true at the same time are called simultaneous equations. (“Simultaneous” means “at the same time”.)
The picture shows the idea of elimination. The two baskets are the same except that the first has one extra pen, and it costs £2 more. So one pen costs £2, and then the book costs £1. We got rid of the book by comparing the baskets: that is elimination. In algebra we do this by adding or subtracting the two equations. This is allowed because if you add equal amounts to equal amounts, the results are equal.
The method.
- Number the equations (1) and (2). Make sure each has the form (number)x + (number)y = (number).
- Choose a letter to eliminate. Make the numbers in front of it the same size in both equations. If they are not, multiply one or both equations (multiply every term, including the right-hand side).
- If the matching terms have the same sign, subtract the equations. If they have different signs, add them. One letter disappears.
- Solve the equation that is left.
- Substitute the value into one of the original equations to find the other letter.
- Check in the other original equation.
A memory aid: Same Signs Subtract. If the equations are not lined up, rearrange them first: 2x = 13 − y becomes 2x + y = 13.
Key ideas
- Simultaneous equations: two equations, two letters, both true at once. The solution is a pair of values.
- Eliminate a letter by adding or subtracting the equations. Same signs: subtract. Different signs: add.
- If no letter has matching numbers, multiply one or both equations first. Multiply every term.
- Find one letter, substitute to find the other, and always check in the other equation.
- Line up the equations first: letters on the left, numbers on the right.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 2y = 12
(2) 3x − y = 3 - Choose a letter to eliminate: x. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of x are 3 and 3: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The x terms are 3x and 3x. They have the same sign, so subtract the equations: 3x − 3x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) 3x + 2y = 12
(2) 3x − y = 3
x terms: 3x − 3x = 0 (they cancel)
y terms: 2y − (−y) = 3y
Numbers: 12 − 3 = 9
3y = 9 - Solve for y. 3y means 3 multiplied by y, so divide both sides by 3.Divide both sides by 3: y = 9 ÷ 3 = 3
- Substitute y = 3 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 3x + 2y = 12
3x + 2 × 3 = 12
3x + 6 = 12
Subtract 6 from both sides: 3x = 6
Divide both sides by 3: x = 6 ÷ 3 = 2 - Write the solution. Give both values. They belong together as a pair.x = 2, y = 3
Check. Put x = 2, y = 3 into the other equation (equation (2), which we did not use to find x): 3 × 2 − 3 = 6 − 3 = 3. This equals 3, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 3y = 12
(2) 4x − 3y = 6 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 3 and −3: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are 3y and −3y. They have different signs, so add the equations: 3y + (−3y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 2x + 3y = 12
(2) 4x − 3y = 6
y terms: 3y + (−3y) = 0 (they cancel)
x terms: 2x + 4x = 6x
Numbers: 12 + 6 = 18
6x = 18 - Solve for x. 6x means 6 multiplied by x, so divide both sides by 6.Divide both sides by 6: x = 18 ÷ 6 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 3y = 12
2 × 3 + 3y = 12
3y + 6 = 12
Subtract 6 from both sides: 3y = 6
Divide both sides by 3: y = 6 ÷ 3 = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 4 × 3 − 3 × 2 = 12 − 6 = 6. This equals 6, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + y = 7
(2) 3x − 2y = 7 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 1 and 2. The lowest common multiple of 1 and 2 is 2, so we will make both equal to 2. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 2x, 2 × y on the left and 2 × 7 on the right.
(3) 4x + 2y = 14 - Add (3) and (2). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 4x + 2y = 14
(2) 3x − 2y = 7
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 4x + 3x = 7x
Numbers: 14 + 7 = 21
7x = 21 - Solve for x. 7x means 7 multiplied by x, so divide both sides by 7.Divide both sides by 7: x = 21 ÷ 7 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + y = 7
2 × 3 + y = 7
y + 6 = 7
Subtract 6 from both sides: y = 1 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 1
Check. Put x = 3, y = 1 into the other equation (equation (2), which we did not use to find y): 3 × 3 − 2 × 1 = 9 − 2 = 7. This equals 7, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 2y = 16
(2) 5x − 3y = 14 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 3. The lowest common multiple of 2 and 3 is 6, so we will make both equal to 6. So multiply equation (1) by 3 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 3x, 3 × 2y on the left and 3 × 16 on the right.
(3) 9x + 6y = 48
Equation (2) × 2: 2 × 5x, 2 × (−3y) on the left and 2 × 14 on the right.
(4) 10x − 6y = 28 - Add (3) and (4). The y terms are 6y and −6y. They have different signs, so add the equations: 6y + (−6y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 9x + 6y = 48
(4) 10x − 6y = 28
y terms: 6y + (−6y) = 0 (they cancel)
x terms: 9x + 10x = 19x
Numbers: 48 + 28 = 76
19x = 76 - Solve for x. 19x means 19 multiplied by x, so divide both sides by 19.Divide both sides by 19: x = 76 ÷ 19 = 4
- Substitute x = 4 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 3x + 2y = 16
3 × 4 + 2y = 16
2y + 12 = 16
Subtract 12 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 2
Check. Put x = 4, y = 2 into the other equation (equation (2), which we did not use to find y): 5 × 4 − 3 × 2 = 20 − 6 = 14. This equals 14, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Rearrange each equation so the letters are on the left and the plain number is on the right. Elimination needs the equations lined up in the same pattern: (number)x + (number)y = (number). Use inverse operations on both sides.2x = 13 − y → 2x + y = 13
x + 3y = 14 → x + 3y = 14 - Number the equations. Call the tidied equations (1) and (2).(1) 2x + y = 13
(2) x + 3y = 14 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 1. The lowest common multiple of 2 and 1 is 2, so we will make both equal to 2. So multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (2) × 2: 2 × x, 2 × 3y on the left and 2 × 14 on the right.
(3) 2x + 6y = 28 - Subtract (1) from (3). The x terms are 2x and 2x. They have the same sign, so subtract the equations: 2x − 2x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(3) 2x + 6y = 28
(1) 2x + y = 13
x terms: 2x − 2x = 0 (they cancel)
y terms: 6y − y = 5y
Numbers: 28 − 13 = 15
5y = 15 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = 15 ÷ 5 = 3
- Substitute y = 3 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): x + 3y = 14
x + 3 × 3 = 14
x + 9 = 14
Subtract 9 from both sides: x = 5 - Write the solution. Give both values. They belong together as a pair.x = 5, y = 3
Check. Put x = 5, y = 3 into the other equation (equation (1), which we did not use to find x): 2 × 5 + 3 = 10 + 3 = 13. This equals 13, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 4y = 5
(2) 6x − 4y = 3 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 4 and −4: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are 4y and −4y. They have different signs, so add the equations: 4y + (−4y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 2x + 4y = 5
(2) 6x − 4y = 3
y terms: 4y + (−4y) = 0 (they cancel)
x terms: 2x + 6x = 8x
Numbers: 5 + 3 = 8
8x = 8 - Solve for x. 8x means 8 multiplied by x, so divide both sides by 8.Divide both sides by 8: x = 8 ÷ 8 = 1
- Substitute x = 1 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 4y = 5
2 × 1 + 4y = 5
4y + 2 = 5
Subtract 2 from both sides: 4y = 3
Divide both sides by 4: y = 3 ÷ 4 = 34 - Write the solution. Give both values. They belong together as a pair.x = 1, y = 34
Check. Put x = 1, y = 34 into the other equation (equation (2), which we did not use to find y): 6 × 1 − 4 × 34 = 6 − 3 = 3. This equals 3, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 2y = 1
(2) 2x + 3y = −1 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 3 and 2. The lowest common multiple of 3 and 2 is 6, so we will make both equal to 6. So multiply equation (1) by 2 and multiply equation (2) by 3. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 3x, 2 × 2y on the left and 2 × 1 on the right.
(3) 6x + 4y = 2
Equation (2) × 3: 3 × 2x, 3 × 3y on the left and 3 × (−1) on the right.
(4) 6x + 9y = −3 - Subtract (3) from (4). The x terms are 6x and 6x. They have the same sign, so subtract the equations: 6x − 6x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(4) 6x + 9y = −3
(3) 6x + 4y = 2
x terms: 6x − 6x = 0 (they cancel)
y terms: 9y − 4y = 5y
Numbers: −3 − 2 = −5
5y = −5 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = −5 ÷ 5 = −1
- Substitute y = −1 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): 2x + 3y = −1
2x + 3 × (−1) = −1
2x − 3 = −1
Add 3 to both sides: 2x = 2
Divide both sides by 2: x = 2 ÷ 2 = 1 - Write the solution. Give both values. They belong together as a pair.x = 1, y = −1
Check. Put x = 1, y = −1 into the other equation (equation (1), which we did not use to find x): 3 × 1 + 2 × (−1) = 3 − 2 = 1. This equals 1, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Watch out
When you subtract, you subtract every term, and subtracting a negative number is the same as adding. For example (2y) − (−y) = 3y, not y. A very common mistake is to subtract only the first term, or to forget to multiply the number on the right-hand side when you multiply an equation.Exercise 14E Simultaneous equations: elimination
Non-calculatorKey words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) x + y = 9
(2) x − y = 3 - Choose a letter to eliminate: x. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of x are 1 and 1: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The x terms are x and x. They have the same sign, so subtract the equations: x − x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) x + y = 9
(2) x − y = 3
x terms: x − x = 0 (they cancel)
y terms: y − (−y) = 2y
Numbers: 9 − 3 = 6
2y = 6 - Solve for y. 2y means 2 multiplied by y, so divide both sides by 2.Divide both sides by 2: y = 6 ÷ 2 = 3
- Substitute y = 3 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + y = 9
x + 3 = 9
x + 3 = 9
Subtract 3 from both sides: x = 6 - Write the solution. Give both values. They belong together as a pair.x = 6, y = 3
Check. Put x = 6, y = 3 into the other equation (equation (2), which we did not use to find x): 6 − 3 = 6 − 3 = 3. This equals 3, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + y = 11
(2) x + y = 7 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 1 and 1: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The y terms are y and y. They have the same sign, so subtract the equations: y − y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 2x + y = 11
(2) x + y = 7
y terms: y − y = 0 (they cancel)
x terms: 2x − x = x
Numbers: 11 − 7 = 4
x = 4 - Read off x. x is already on its own.x = 4
- Substitute x = 4 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + y = 11
2 × 4 + y = 11
y + 8 = 11
Subtract 8 from both sides: y = 3 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 3
Check. Put x = 4, y = 3 into the other equation (equation (2), which we did not use to find y): 4 + 3 = 4 + 3 = 7. This equals 7, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + y = 14
(2) x + y = 8 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 1 and 1: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The y terms are y and y. They have the same sign, so subtract the equations: y − y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 3x + y = 14
(2) x + y = 8
y terms: y − y = 0 (they cancel)
x terms: 3x − x = 2x
Numbers: 14 − 8 = 6
2x = 6 - Solve for x. 2x means 2 multiplied by x, so divide both sides by 2.Divide both sides by 2: x = 6 ÷ 2 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 3x + y = 14
3 × 3 + y = 14
y + 9 = 14
Subtract 9 from both sides: y = 5 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 5
Check. Put x = 3, y = 5 into the other equation (equation (2), which we did not use to find y): 3 + 5 = 3 + 5 = 8. This equals 8, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) x + 2y = 9
(2) x + y = 6 - Choose a letter to eliminate: x. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of x are 1 and 1: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The x terms are x and x. They have the same sign, so subtract the equations: x − x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) x + 2y = 9
(2) x + y = 6
x terms: x − x = 0 (they cancel)
y terms: 2y − y = y
Numbers: 9 − 6 = 3
y = 3 - Read off y. y is already on its own.y = 3
- Substitute y = 3 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + 2y = 9
x + 2 × 3 = 9
x + 6 = 9
Subtract 6 from both sides: x = 3 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 3
Check. Put x = 3, y = 3 into the other equation (equation (2), which we did not use to find x): 3 + 3 = 3 + 3 = 6. This equals 6, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 4x + 3y = 27
(2) 4x − y = 7 - Choose a letter to eliminate: x. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of x are 4 and 4: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The x terms are 4x and 4x. They have the same sign, so subtract the equations: 4x − 4x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) 4x + 3y = 27
(2) 4x − y = 7
x terms: 4x − 4x = 0 (they cancel)
y terms: 3y − (−y) = 4y
Numbers: 27 − 7 = 20
4y = 20 - Solve for y. 4y means 4 multiplied by y, so divide both sides by 4.Divide both sides by 4: y = 20 ÷ 4 = 5
- Substitute y = 5 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 4x + 3y = 27
4x + 3 × 5 = 27
4x + 15 = 27
Subtract 15 from both sides: 4x = 12
Divide both sides by 4: x = 12 ÷ 4 = 3 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 5
Check. Put x = 3, y = 5 into the other equation (equation (2), which we did not use to find x): 4 × 3 − 5 = 12 − 5 = 7. This equals 7, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 5x + 2y = 24
(2) 3x − 2y = 8 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 2 and −2: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 5x + 2y = 24
(2) 3x − 2y = 8
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 5x + 3x = 8x
Numbers: 24 + 8 = 32
8x = 32 - Solve for x. 8x means 8 multiplied by x, so divide both sides by 8.Divide both sides by 8: x = 32 ÷ 8 = 4
- Substitute x = 4 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 5x + 2y = 24
5 × 4 + 2y = 24
2y + 20 = 24
Subtract 20 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 2
Check. Put x = 4, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 4 − 2 × 2 = 12 − 4 = 8. This equals 8, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
2x + 3y = 12
2x + y = 8
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 3y = 12
(2) 2x + y = 8 - Choose a letter to eliminate: x. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of x are 2 and 2: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The x terms are 2x and 2x. They have the same sign, so subtract the equations: 2x − 2x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) 2x + 3y = 12
(2) 2x + y = 8
x terms: 2x − 2x = 0 (they cancel)
y terms: 3y − y = 2y
Numbers: 12 − 8 = 4
2y = 4 - Solve for y. 2y means 2 multiplied by y, so divide both sides by 2.Divide both sides by 2: y = 4 ÷ 2 = 2
- Substitute y = 2 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 3y = 12
2x + 3 × 2 = 12
2x + 6 = 12
Subtract 6 from both sides: 2x = 6
Divide both sides by 2: x = 6 ÷ 2 = 3 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find x): 2 × 3 + 2 = 6 + 2 = 8. This equals 8, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
5x + 2y = 19
3x − 2y = 5
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 5x + 2y = 19
(2) 3x − 2y = 5 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 2 and −2: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 5x + 2y = 19
(2) 3x − 2y = 5
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 5x + 3x = 8x
Numbers: 19 + 5 = 24
8x = 24 - Solve for x. 8x means 8 multiplied by x, so divide both sides by 8.Divide both sides by 8: x = 24 ÷ 8 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 5x + 2y = 19
5 × 3 + 2y = 19
2y + 15 = 19
Subtract 15 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 3 − 2 × 2 = 9 − 4 = 5. This equals 5, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + y = 8
(2) 3x − 2y = 5 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 1 and 2. The lowest common multiple of 1 and 2 is 2, so we will make both equal to 2. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 2x, 2 × y on the left and 2 × 8 on the right.
(3) 4x + 2y = 16 - Add (3) and (2). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 4x + 2y = 16
(2) 3x − 2y = 5
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 4x + 3x = 7x
Numbers: 16 + 5 = 21
7x = 21 - Solve for x. 7x means 7 multiplied by x, so divide both sides by 7.Divide both sides by 7: x = 21 ÷ 7 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + y = 8
2 × 3 + y = 8
y + 6 = 8
Subtract 6 from both sides: y = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 3 − 2 × 2 = 9 − 4 = 5. This equals 5, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) x + 3y = 11
(2) 2x − y = 1 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 1 and 2. The lowest common multiple of 1 and 2 is 2, so we will make both equal to 2. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × x, 2 × 3y on the left and 2 × 11 on the right.
(3) 2x + 6y = 22 - Subtract (2) from (3). The x terms are 2x and 2x. They have the same sign, so subtract the equations: 2x − 2x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(3) 2x + 6y = 22
(2) 2x − y = 1
x terms: 2x − 2x = 0 (they cancel)
y terms: 6y − (−y) = 7y
Numbers: 22 − 1 = 21
7y = 21 - Solve for y. 7y means 7 multiplied by y, so divide both sides by 7.Divide both sides by 7: y = 21 ÷ 7 = 3
- Substitute y = 3 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + 3y = 11
x + 3 × 3 = 11
x + 9 = 11
Subtract 9 from both sides: x = 2 - Write the solution. Give both values. They belong together as a pair.x = 2, y = 3
Check. Put x = 2, y = 3 into the other equation (equation (2), which we did not use to find x): 2 × 2 − 3 = 4 − 3 = 1. This equals 1, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 2y = 14
(2) x − y = 3 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 1. The lowest common multiple of 2 and 1 is 2, so we will make both equal to 2. So multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (2) × 2: 2 × x, 2 × (−y) on the left and 2 × 3 on the right.
(3) 2x − 2y = 6 - Add (1) and (3). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 3x + 2y = 14
(3) 2x − 2y = 6
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 3x + 2x = 5x
Numbers: 14 + 6 = 20
5x = 20 - Solve for x. 5x means 5 multiplied by x, so divide both sides by 5.Divide both sides by 5: x = 20 ÷ 5 = 4
- Substitute x = 4 into equation (2) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): x − y = 3
4 − y = 3
−y + 4 = 3
Subtract 4 from both sides: −y = −1
Divide both sides by −1: y = −1 ÷ (−1) = 1 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 1
Check. Put x = 4, y = 1 into the other equation (equation (1), which we did not use to find y): 3 × 4 + 2 × 1 = 12 + 2 = 14. This equals 14, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 5x − y = 13
(2) 2x + 3y = 12 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 1 and 3. The lowest common multiple of 1 and 3 is 3, so we will make both equal to 3. So multiply equation (1) by 3. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 5x, 3 × (−y) on the left and 3 × 13 on the right.
(3) 15x − 3y = 39 - Add (3) and (2). The y terms are −3y and 3y. They have different signs, so add the equations: −3y + 3y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 15x − 3y = 39
(2) 2x + 3y = 12
y terms: −3y + 3y = 0 (they cancel)
x terms: 15x + 2x = 17x
Numbers: 39 + 12 = 51
17x = 51 - Solve for x. 17x means 17 multiplied by x, so divide both sides by 17.Divide both sides by 17: x = 51 ÷ 17 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 5x − y = 13
5 × 3 − y = 13
−y + 15 = 13
Subtract 15 from both sides: −y = −2
Divide both sides by −1: y = −2 ÷ (−1) = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 2 × 3 + 3 × 2 = 6 + 6 = 12. This equals 12, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) x + 2y = 8
(2) 3x + 4y = 20 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 4. The lowest common multiple of 2 and 4 is 4, so we will make both equal to 4. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × x, 2 × 2y on the left and 2 × 8 on the right.
(3) 2x + 4y = 16 - Subtract (3) from (2). The y terms are 4y and 4y. They have the same sign, so subtract the equations: 4y − 4y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(2) 3x + 4y = 20
(3) 2x + 4y = 16
y terms: 4y − 4y = 0 (they cancel)
x terms: 3x − 2x = x
Numbers: 20 − 16 = 4
x = 4 - Read off x. x is already on its own.x = 4
- Substitute x = 4 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + 2y = 8
4 + 2y = 8
2y + 4 = 8
Subtract 4 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 2
Check. Put x = 4, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 4 + 4 × 2 = 12 + 8 = 20. This equals 20, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 4x + y = 14
(2) 3x − 2y = 5 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 1 and 2. The lowest common multiple of 1 and 2 is 2, so we will make both equal to 2. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 4x, 2 × y on the left and 2 × 14 on the right.
(3) 8x + 2y = 28 - Add (3) and (2). The y terms are 2y and −2y. They have different signs, so add the equations: 2y + (−2y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 8x + 2y = 28
(2) 3x − 2y = 5
y terms: 2y + (−2y) = 0 (they cancel)
x terms: 8x + 3x = 11x
Numbers: 28 + 5 = 33
11x = 33 - Solve for x. 11x means 11 multiplied by x, so divide both sides by 11.Divide both sides by 11: x = 33 ÷ 11 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 4x + y = 14
4 × 3 + y = 14
y + 12 = 14
Subtract 12 from both sides: y = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 3 − 2 × 2 = 9 − 4 = 5. This equals 5, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 3y = 13
(2) 3x + 2y = 12 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 3. The lowest common multiple of 2 and 3 is 6, so we will make both equal to 6. So multiply equation (1) by 3 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 2x, 3 × 3y on the left and 3 × 13 on the right.
(3) 6x + 9y = 39
Equation (2) × 2: 2 × 3x, 2 × 2y on the left and 2 × 12 on the right.
(4) 6x + 4y = 24 - Subtract (4) from (3). The x terms are 6x and 6x. They have the same sign, so subtract the equations: 6x − 6x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(3) 6x + 9y = 39
(4) 6x + 4y = 24
x terms: 6x − 6x = 0 (they cancel)
y terms: 9y − 4y = 5y
Numbers: 39 − 24 = 15
5y = 15 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = 15 ÷ 5 = 3
- Substitute y = 3 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 3y = 13
2x + 3 × 3 = 13
2x + 9 = 13
Subtract 9 from both sides: 2x = 4
Divide both sides by 2: x = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 2, y = 3
Check. Put x = 2, y = 3 into the other equation (equation (2), which we did not use to find x): 3 × 2 + 2 × 3 = 6 + 6 = 12. This equals 12, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 4y = 24
(2) 2x + 3y = 17 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 3 and 2. The lowest common multiple of 3 and 2 is 6, so we will make both equal to 6. So multiply equation (1) by 2 and multiply equation (2) by 3. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 3x, 2 × 4y on the left and 2 × 24 on the right.
(3) 6x + 8y = 48
Equation (2) × 3: 3 × 2x, 3 × 3y on the left and 3 × 17 on the right.
(4) 6x + 9y = 51 - Subtract (3) from (4). The x terms are 6x and 6x. They have the same sign, so subtract the equations: 6x − 6x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(4) 6x + 9y = 51
(3) 6x + 8y = 48
x terms: 6x − 6x = 0 (they cancel)
y terms: 9y − 8y = y
Numbers: 51 − 48 = 3
y = 3 - Read off y. y is already on its own.y = 3
- Substitute y = 3 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): 2x + 3y = 17
2x + 3 × 3 = 17
2x + 9 = 17
Subtract 9 from both sides: 2x = 8
Divide both sides by 2: x = 8 ÷ 2 = 4 - Write the solution. Give both values. They belong together as a pair.x = 4, y = 3
Check. Put x = 4, y = 3 into the other equation (equation (1), which we did not use to find x): 3 × 4 + 4 × 3 = 12 + 12 = 24. This equals 24, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 5x + 2y = 19
(2) 3x + 5y = 19 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 5. The lowest common multiple of 2 and 5 is 10, so we will make both equal to 10. So multiply equation (1) by 5 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 5: 5 × 5x, 5 × 2y on the left and 5 × 19 on the right.
(3) 25x + 10y = 95
Equation (2) × 2: 2 × 3x, 2 × 5y on the left and 2 × 19 on the right.
(4) 6x + 10y = 38 - Subtract (4) from (3). The y terms are 10y and 10y. They have the same sign, so subtract the equations: 10y − 10y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 25x + 10y = 95
(4) 6x + 10y = 38
y terms: 10y − 10y = 0 (they cancel)
x terms: 25x − 6x = 19x
Numbers: 95 − 38 = 57
19x = 57 - Solve for x. 19x means 19 multiplied by x, so divide both sides by 19.Divide both sides by 19: x = 57 ÷ 19 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 5x + 2y = 19
5 × 3 + 2y = 19
2y + 15 = 19
Subtract 15 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 × 3 + 5 × 2 = 9 + 10 = 19. This equals 19, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 4x − 3y = 6
(2) 3x + 2y = 13 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 3 and 2. The lowest common multiple of 3 and 2 is 6, so we will make both equal to 6. So multiply equation (1) by 2 and multiply equation (2) by 3. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 4x, 2 × (−3y) on the left and 2 × 6 on the right.
(3) 8x − 6y = 12
Equation (2) × 3: 3 × 3x, 3 × 2y on the left and 3 × 13 on the right.
(4) 9x + 6y = 39 - Add (3) and (4). The y terms are −6y and 6y. They have different signs, so add the equations: −6y + 6y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 8x − 6y = 12
(4) 9x + 6y = 39
y terms: −6y + 6y = 0 (they cancel)
x terms: 8x + 9x = 17x
Numbers: 12 + 39 = 51
17x = 51 - Solve for x. 17x means 17 multiplied by x, so divide both sides by 17.Divide both sides by 17: x = 51 ÷ 17 = 3
- Substitute x = 3 into equation (2) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): 3x + 2y = 13
3 × 3 + 2y = 13
2y + 9 = 13
Subtract 9 from both sides: 2y = 4
Divide both sides by 2: y = 4 ÷ 2 = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (1), which we did not use to find y): 4 × 3 − 3 × 2 = 12 − 6 = 6. This equals 6, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 7x + 2y = 20
(2) 3x − 5y = −9 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 5. The lowest common multiple of 2 and 5 is 10, so we will make both equal to 10. So multiply equation (1) by 5 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 5: 5 × 7x, 5 × 2y on the left and 5 × 20 on the right.
(3) 35x + 10y = 100
Equation (2) × 2: 2 × 3x, 2 × (−5y) on the left and 2 × (−9) on the right.
(4) 6x − 10y = −18 - Add (3) and (4). The y terms are 10y and −10y. They have different signs, so add the equations: 10y + (−10y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 35x + 10y = 100
(4) 6x − 10y = −18
y terms: 10y + (−10y) = 0 (they cancel)
x terms: 35x + 6x = 41x
Numbers: 100 + (−18) = 82
41x = 82 - Solve for x. 41x means 41 multiplied by x, so divide both sides by 41.Divide both sides by 41: x = 82 ÷ 41 = 2
- Substitute x = 2 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 7x + 2y = 20
7 × 2 + 2y = 20
2y + 14 = 20
Subtract 14 from both sides: 2y = 6
Divide both sides by 2: y = 6 ÷ 2 = 3 - Write the solution. Give both values. They belong together as a pair.x = 2, y = 3
Check. Put x = 2, y = 3 into the other equation (equation (2), which we did not use to find y): 3 × 2 − 5 × 3 = 6 − 15 = −9. This equals −9, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x − 5y = −4
(2) 3x + 2y = 13 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 3. The lowest common multiple of 2 and 3 is 6, so we will make both equal to 6. So multiply equation (1) by 3 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 2x, 3 × (−5y) on the left and 3 × (−4) on the right.
(3) 6x − 15y = −12
Equation (2) × 2: 2 × 3x, 2 × 2y on the left and 2 × 13 on the right.
(4) 6x + 4y = 26 - Subtract (3) from (4). The x terms are 6x and 6x. They have the same sign, so subtract the equations: 6x − 6x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(4) 6x + 4y = 26
(3) 6x − 15y = −12
x terms: 6x − 6x = 0 (they cancel)
y terms: 4y − (−15y) = 19y
Numbers: 26 − (−12) = 38
19y = 38 - Solve for y. 19y means 19 multiplied by y, so divide both sides by 19.Divide both sides by 19: y = 38 ÷ 19 = 2
- Substitute y = 2 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x − 5y = −4
2x − 5 × 2 = −4
2x − 10 = −4
Add 10 to both sides: 2x = 6
Divide both sides by 2: x = 6 ÷ 2 = 3 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find x): 3 × 3 + 2 × 2 = 9 + 4 = 13. This equals 13, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
x + y = 10
3x + y = 22
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) x + y = 10
(2) 3x + y = 22 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 1 and 1: they are already the same size, so no multiplying is needed.
- Subtract (1) from (2). The y terms are y and y. They have the same sign, so subtract the equations: y − y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(2) 3x + y = 22
(1) x + y = 10
y terms: y − y = 0 (they cancel)
x terms: 3x − x = 2x
Numbers: 22 − 10 = 12
2x = 12 - Solve for x. 2x means 2 multiplied by x, so divide both sides by 2.Divide both sides by 2: x = 12 ÷ 2 = 6
- Substitute x = 6 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + y = 10
6 + y = 10
y + 6 = 10
Subtract 6 from both sides: y = 4 - Write the solution. Give both values. They belong together as a pair.x = 6, y = 4
Check. Put x = 6, y = 4 into the other equation (equation (2), which we did not use to find y): 3 × 6 + 4 = 18 + 4 = 22. This equals 22, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
4x + 3y = 18
x + 3y = 9
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 4x + 3y = 18
(2) x + 3y = 9 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 3 and 3: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The y terms are 3y and 3y. They have the same sign, so subtract the equations: 3y − 3y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 4x + 3y = 18
(2) x + 3y = 9
y terms: 3y − 3y = 0 (they cancel)
x terms: 4x − x = 3x
Numbers: 18 − 9 = 9
3x = 9 - Solve for x. 3x means 3 multiplied by x, so divide both sides by 3.Divide both sides by 3: x = 9 ÷ 3 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 4x + 3y = 18
4 × 3 + 3y = 18
3y + 12 = 18
Subtract 12 from both sides: 3y = 6
Divide both sides by 3: y = 6 ÷ 3 = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 + 3 × 2 = 3 + 6 = 9. This equals 9, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Rearrange each equation so the letters are on the left and the plain number is on the right. Elimination needs the equations lined up in the same pattern: (number)x + (number)y = (number). Use inverse operations on both sides.2x = 13 − y → 2x + y = 13
x + 3y = 14 → x + 3y = 14 - Number the equations. Call the tidied equations (1) and (2).(1) 2x + y = 13
(2) x + 3y = 14 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 1. The lowest common multiple of 2 and 1 is 2, so we will make both equal to 2. So multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (2) × 2: 2 × x, 2 × 3y on the left and 2 × 14 on the right.
(3) 2x + 6y = 28 - Subtract (1) from (3). The x terms are 2x and 2x. They have the same sign, so subtract the equations: 2x − 2x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(3) 2x + 6y = 28
(1) 2x + y = 13
x terms: 2x − 2x = 0 (they cancel)
y terms: 6y − y = 5y
Numbers: 28 − 13 = 15
5y = 15 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = 15 ÷ 5 = 3
- Substitute y = 3 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): x + 3y = 14
x + 3 × 3 = 14
x + 9 = 14
Subtract 9 from both sides: x = 5 - Write the solution. Give both values. They belong together as a pair.x = 5, y = 3
Check. Put x = 5, y = 3 into the other equation (equation (1), which we did not use to find x): 2 × 5 + 3 = 10 + 3 = 13. This equals 13, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Rearrange each equation so the letters are on the left and the plain number is on the right. Elimination needs the equations lined up in the same pattern: (number)x + (number)y = (number). Use inverse operations on both sides.y = 8 − x → x + y = 8
3x − y = 4 → 3x − y = 4 - Number the equations. Call the tidied equations (1) and (2).(1) x + y = 8
(2) 3x − y = 4 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 1 and −1: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are y and −y. They have different signs, so add the equations: y + (−y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) x + y = 8
(2) 3x − y = 4
y terms: y + (−y) = 0 (they cancel)
x terms: x + 3x = 4x
Numbers: 8 + 4 = 12
4x = 12 - Solve for x. 4x means 4 multiplied by x, so divide both sides by 4.Divide both sides by 4: x = 12 ÷ 4 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): x + y = 8
3 + y = 8
y + 3 = 8
Subtract 3 from both sides: y = 5 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 5
Check. Put x = 3, y = 5 into the other equation (equation (2), which we did not use to find y): 3 × 3 − 5 = 9 − 5 = 4. This equals 4, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Rearrange each equation so the letters are on the left and the plain number is on the right. Elimination needs the equations lined up in the same pattern: (number)x + (number)y = (number). Use inverse operations on both sides.2x + 3y = 19 → 2x + 3y = 19
x = y + 2 → x − y = 2 - Number the equations. Call the tidied equations (1) and (2).(1) 2x + 3y = 19
(2) x − y = 2 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 1. The lowest common multiple of 2 and 1 is 2, so we will make both equal to 2. So multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (2) × 2: 2 × x, 2 × (−y) on the left and 2 × 2 on the right.
(3) 2x − 2y = 4 - Subtract (3) from (1). The x terms are 2x and 2x. They have the same sign, so subtract the equations: 2x − 2x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(1) 2x + 3y = 19
(3) 2x − 2y = 4
x terms: 2x − 2x = 0 (they cancel)
y terms: 3y − (−2y) = 5y
Numbers: 19 − 4 = 15
5y = 15 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = 15 ÷ 5 = 3
- Substitute y = 3 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): x − y = 2
x − 3 = 2
x − 3 = 2
Add 3 to both sides: x = 5 - Write the solution. Give both values. They belong together as a pair.x = 5, y = 3
Check. Put x = 5, y = 3 into the other equation (equation (1), which we did not use to find x): 2 × 5 + 3 × 3 = 10 + 9 = 19. This equals 19, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Rearrange each equation so the letters are on the left and the plain number is on the right. Elimination needs the equations lined up in the same pattern: (number)x + (number)y = (number). Use inverse operations on both sides.3x = 11 − y → 3x + y = 11
x + 2y = 7 → x + 2y = 7 - Number the equations. Call the tidied equations (1) and (2).(1) 3x + y = 11
(2) x + 2y = 7 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 1 and 2. The lowest common multiple of 1 and 2 is 2, so we will make both equal to 2. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 3x, 2 × y on the left and 2 × 11 on the right.
(3) 6x + 2y = 22 - Subtract (2) from (3). The y terms are 2y and 2y. They have the same sign, so subtract the equations: 2y − 2y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 6x + 2y = 22
(2) x + 2y = 7
y terms: 2y − 2y = 0 (they cancel)
x terms: 6x − x = 5x
Numbers: 22 − 7 = 15
5x = 15 - Solve for x. 5x means 5 multiplied by x, so divide both sides by 5.Divide both sides by 5: x = 15 ÷ 5 = 3
- Substitute x = 3 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 3x + y = 11
3 × 3 + y = 11
y + 9 = 11
Subtract 9 from both sides: y = 2 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (2), which we did not use to find y): 3 + 2 × 2 = 3 + 4 = 7. This equals 7, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Method. Adding the equations does not remove a letter here, because the signs are the same. Subtract instead.
- Try Maya's idea: add the equations. Adding the x terms gives 4x + x = 5x and the y terms give 3y + 3y = 6y. Neither letter has gone, so adding has not helped.5x + 6y = 27
- Look at the signs. The y terms are +3y and +3y. They have the same sign, so we should subtract.Same signs: subtract.
- Subtract (2) from (1). 4x − x = 3x, 3y − 3y = 0 and 18 − 9 = 9.3x = 9, so x = 3
- Find y. Put x = 3 into equation (2): 3 + 3y = 9.3y = 6, so y = 2
Check. (1): 4 × 3 + 3 × 2 = 12 + 6 = 18. ✓ (2): 3 + 3 × 2 = 9. ✓
Watch out. Same signs subtract, different signs add.
Method. Try to eliminate a letter and see what happens.
- Number the equations. Call them (1) and (2).(1) x + y = 5
(2) x + y = 7 - Subtract (1) from (2). Both letters disappear at once.(x + y) − (x + y) = 7 − 5
0 = 2 - Explain the result. 0 is not equal to 2, so the equations contradict each other. The same two numbers cannot add to make both 5 and 7.There is no solution.
Check. Try some pairs: if x + y = 5 then x + y is not 7, whatever x and y are. ✓
Watch out. If both letters vanish and you are left with a false statement, there is no solution. (The two lines would be parallel.)
3x + 5y = 21
2x + 5y = 19
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 5y = 21
(2) 2x + 5y = 19 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 5 and 5: they are already the same size, so no multiplying is needed.
- Subtract (2) from (1). The y terms are 5y and 5y. They have the same sign, so subtract the equations: 5y − 5y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 3x + 5y = 21
(2) 2x + 5y = 19
y terms: 5y − 5y = 0 (they cancel)
x terms: 3x − 2x = x
Numbers: 21 − 19 = 2
x = 2 - Read off x. x is already on its own.x = 2
- Substitute x = 2 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 3x + 5y = 21
3 × 2 + 5y = 21
5y + 6 = 21
Subtract 6 from both sides: 5y = 15
Divide both sides by 5: y = 15 ÷ 5 = 3 - Write the solution. Give both values. They belong together as a pair.x = 2, y = 3
Check. Put x = 2, y = 3 into the other equation (equation (2), which we did not use to find y): 2 × 2 + 5 × 3 = 4 + 15 = 19. This equals 19, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 3y = −1
(2) 3x + 2y = 1 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 2 and 3. The lowest common multiple of 2 and 3 is 6, so we will make both equal to 6. So multiply equation (1) by 3 and multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 2x, 3 × 3y on the left and 3 × (−1) on the right.
(3) 6x + 9y = −3
Equation (2) × 2: 2 × 3x, 2 × 2y on the left and 2 × 1 on the right.
(4) 6x + 4y = 2 - Subtract (4) from (3). The x terms are 6x and 6x. They have the same sign, so subtract the equations: 6x − 6x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(3) 6x + 9y = −3
(4) 6x + 4y = 2
x terms: 6x − 6x = 0 (they cancel)
y terms: 9y − 4y = 5y
Numbers: −3 − 2 = −5
5y = −5 - Solve for y. 5y means 5 multiplied by y, so divide both sides by 5.Divide both sides by 5: y = −5 ÷ 5 = −1
- Substitute y = −1 into equation (1) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 3y = −1
2x + 3 × (−1) = −1
2x − 3 = −1
Add 3 to both sides: 2x = 2
Divide both sides by 2: x = 2 ÷ 2 = 1 - Write the solution. Give both values. They belong together as a pair.x = 1, y = −1
Check. Put x = 1, y = −1 into the other equation (equation (2), which we did not use to find x): 3 × 1 + 2 × (−1) = 3 − 2 = 1. This equals 1, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 4x − 5y = 2
(2) 3x + 2y = 13 - Choose a letter to eliminate: x, and make the numbers match. The numbers in front of x are 4 and 3. The lowest common multiple of 4 and 3 is 12, so we will make both equal to 12. So multiply equation (1) by 3 and multiply equation (2) by 4. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of x match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 3: 3 × 4x, 3 × (−5y) on the left and 3 × 2 on the right.
(3) 12x − 15y = 6
Equation (2) × 4: 4 × 3x, 4 × 2y on the left and 4 × 13 on the right.
(4) 12x + 8y = 52 - Subtract (3) from (4). The x terms are 12x and 12x. They have the same sign, so subtract the equations: 12x − 12x = 0, so x disappears. Do the same thing to the left sides and to the right sides.(4) 12x + 8y = 52
(3) 12x − 15y = 6
x terms: 12x − 12x = 0 (they cancel)
y terms: 8y − (−15y) = 23y
Numbers: 52 − 6 = 46
23y = 46 - Solve for y. 23y means 23 multiplied by y, so divide both sides by 23.Divide both sides by 23: y = 46 ÷ 23 = 2
- Substitute y = 2 into equation (2) to find x. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): 3x + 2y = 13
3x + 2 × 2 = 13
3x + 4 = 13
Subtract 4 from both sides: 3x = 9
Divide both sides by 3: x = 9 ÷ 3 = 3 - Write the solution. Give both values. They belong together as a pair.x = 3, y = 2
Check. Put x = 3, y = 2 into the other equation (equation (1), which we did not use to find x): 4 × 3 − 5 × 2 = 12 − 10 = 2. This equals 2, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 2x + 4y = 5
(2) 6x − 4y = 3 - Choose a letter to eliminate: y. To eliminate a letter, the numbers in front of it must be the same size in both equations. The numbers in front of y are 4 and −4: they are already the same size, so no multiplying is needed.
- Add (1) and (2). The y terms are 4y and −4y. They have different signs, so add the equations: 4y + (−4y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 2x + 4y = 5
(2) 6x − 4y = 3
y terms: 4y + (−4y) = 0 (they cancel)
x terms: 2x + 6x = 8x
Numbers: 5 + 3 = 8
8x = 8 - Solve for x. 8x means 8 multiplied by x, so divide both sides by 8.Divide both sides by 8: x = 8 ÷ 8 = 1
- Substitute x = 1 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 2x + 4y = 5
2 × 1 + 4y = 5
4y + 2 = 5
Subtract 2 from both sides: 4y = 3
Divide both sides by 4: y = 3 ÷ 4 = 34 - Write the solution. Give both values. They belong together as a pair.x = 1, y = 34
Check. Put x = 1, y = 34 into the other equation (equation (2), which we did not use to find y): 6 × 1 − 4 × 34 = 6 − 3 = 3. This equals 3, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 5x + 2y = 1
(2) 3x + 4y = −5 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 2 and 4. The lowest common multiple of 2 and 4 is 4, so we will make both equal to 4. So multiply equation (1) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (1) × 2: 2 × 5x, 2 × 2y on the left and 2 × 1 on the right.
(3) 10x + 4y = 2 - Subtract (2) from (3). The y terms are 4y and 4y. They have the same sign, so subtract the equations: 4y − 4y = 0, so y disappears. Do the same thing to the left sides and to the right sides.(3) 10x + 4y = 2
(2) 3x + 4y = −5
y terms: 4y − 4y = 0 (they cancel)
x terms: 10x − 3x = 7x
Numbers: 2 − (−5) = 7
7x = 7 - Solve for x. 7x means 7 multiplied by x, so divide both sides by 7.Divide both sides by 7: x = 7 ÷ 7 = 1
- Substitute x = 1 into equation (1) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (1): 5x + 2y = 1
5 × 1 + 2y = 1
2y + 5 = 1
Subtract 5 from both sides: 2y = −4
Divide both sides by 2: y = −4 ÷ 2 = −2 - Write the solution. Give both values. They belong together as a pair.x = 1, y = −2
Check. Put x = 1, y = −2 into the other equation (equation (2), which we did not use to find y): 3 × 1 + 4 × (−2) = 3 − 8 = −5. This equals −5, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
3x + 4y = 2
5x − 2y = 12
Key words. Simultaneous equations are two equations with two unknown letters that must both be true at the same time. To eliminate a letter is to get rid of it. A solution is a pair of values (one for each letter) that makes both equations true.
Method. Simultaneous equations: make the number in front of one letter the same size in both equations, then add or subtract to eliminate (remove) that letter. Solve for the other letter, substitute back, and check.
- Number the equations. We will keep referring to them, so call them (1) and (2).(1) 3x + 4y = 2
(2) 5x − 2y = 12 - Choose a letter to eliminate: y, and make the numbers match. The numbers in front of y are 4 and 2. The lowest common multiple of 4 and 2 is 4, so we will make both equal to 4. So multiply equation (2) by 2. Multiply every term, including the number on the right-hand side.
- Multiply to make the numbers in front of y match. Multiplying every term in an equation by the same number gives an equation that is still true, with the same solution.Equation (2) × 2: 2 × 5x, 2 × (−2y) on the left and 2 × 12 on the right.
(3) 10x − 4y = 24 - Add (1) and (3). The y terms are 4y and −4y. They have different signs, so add the equations: 4y + (−4y) = 0, so y disappears. Do the same thing to the left sides and to the right sides.(1) 3x + 4y = 2
(3) 10x − 4y = 24
y terms: 4y + (−4y) = 0 (they cancel)
x terms: 3x + 10x = 13x
Numbers: 2 + 24 = 26
13x = 26 - Solve for x. 13x means 13 multiplied by x, so divide both sides by 13.Divide both sides by 13: x = 26 ÷ 13 = 2
- Substitute x = 2 into equation (2) to find y. Use one of the original equations (choose the one with the smaller numbers). Replace the letter we know by its value.Equation (2): 5x − 2y = 12
5 × 2 − 2y = 12
−2y + 10 = 12
Subtract 10 from both sides: −2y = 2
Divide both sides by −2: y = 2 ÷ (−2) = −1 - Write the solution. Give both values. They belong together as a pair.x = 2, y = −1
Check. Put x = 2, y = −1 into the other equation (equation (1), which we did not use to find y): 3 × 2 + 4 × (−1) = 6 − 4 = 2. This equals 2, as it should. ✓
Watch out. Same signs subtract, different signs add. Multiply every term of the equation, including the number on the right, and take care with negative numbers when you subtract.
Method. Use the difference of two squares, which you met in section 14.1: x2 − y2 = (x − y)(x + y).
- Write the rule. This lets us use the two sums we know.x2 − y2 = (x − y)(x + y)
- Substitute the known values. We are told x + y = 10 and x − y = 2.= 2 × 10 = 20
- Check by solving. Adding the equations gives 2x = 12, so x = 6 and y = 4.62 − 42 = 36 − 16 = 20
Check. Both routes give 20. ✓
Watch out. You do not always need to find x and y separately.