GCSE Maths Linear Equations (Higher): check-up and exam-style questions

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 9 = 20
2. Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 9 − 9 = 20 − 9
   x = 11
3. Write the solution. x is now on its own, so the equation tells us its value.
   x = 11

Check. Substitute x = 11 into the original equation. Left side: 11 + 9 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   4x = 36
2. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 364
   x = 364 = 9
3. Write the solution. x is now on its own, so the equation tells us its value.
   x = 9

Check. Substitute x = 9 into the original equation. Left side: 4 × 9 = 36. Right side: 36. Both sides equal 36, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   3x + 7 = 25
2. Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   3x + 7 − 7 = 25 − 7
   3x = 18
3. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 183
   x = 183 = 6
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 6

Check. Substitute x = 6 into the original equation. Left side: 3 × 6 + 7 = 18 + 7 = 25. Right side: 25. Both sides equal 25, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   5x − 8 = 12
2. Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   5x − 8 + 8 = 12 + 8
   5x = 20
3. Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.
   5x5 = 205
   x = 205 = 4
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4

Check. Substitute x = 4 into the original equation. Left side: 5 × 4 − 8 = 20 − 8 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   2x + 9 = 3
2. Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   2x + 9 − 9 = 3 − 9
   2x = −6
3. Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.
   2x2 = −62
   x = −62 = −3
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = −3

Check. Substitute x = −3 into the original equation. Left side: 2 × (−3) + 9 = −6 + 9 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   7x − 5 = 3x + 11
2. Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   7x − 5 − 3x = 3x + 11 − 3x
   4x − 5 = 11
3. Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x − 5 + 5 = 11 + 5
   4x = 16
4. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 164
   x = 164 = 4
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4

Check. Substitute x = 4 into the original equation. Left side: 7 × 4 − 5 = 28 − 5 = 23. Right side: 3 × 4 + 11 = 12 + 11 = 23. Both sides equal 23, so the solution is correct. ✓

Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   4(x − 3) = 20
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 4(x − 3):
   4 × x = 4x
   4 × (−3) = −12
   
   The equation is now 4x − 12 = 20
3. Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x − 12 + 12 = 20 + 12
   4x = 32
4. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 324
   x = 324 = 8
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 8

Check. Substitute x = 8 into the original equation. Left side: 4 × (8 − 3) = 4 × 5 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓

Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x + 53 = 4
2. Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.
   3  →  the lowest common multiple is 3
3. Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.
   3 × x + 53 = (x + 5)   (3 goes into 3 exactly 1 time, so the bottom disappears)
   3 × 4 = 12
   The new equation is (x + 5) = 12
4. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
   
   The equation is now x + 5 = 12
5. Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 5 − 5 = 12 − 5
   x = 7
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7

Check. Substitute x = 7 into the original equation. Left side: 7 + 53 = 123 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓

Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.

1. Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.
   Let x = the number Tom thinks of.
2. Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.
   He multiplies it by 3 (3x) and subtracts 4 (3x − 4). The answer is 17.
   Equation: 3x − 4 = 17
3. Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   3x − 4 + 4 = 17 + 4
   3x = 21
4. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   3x3 = 213
   x = 213 = 7
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7
6. Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.
   Tom's number is 7.

Check. 3 × 7 = 21, and 21 − 4 = 17 ✓

Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.

1. Use angles on a straight line. Angles on a straight line add up to 180°.
   (4x + 20) + (2x + 40) = 180
   Equation: (4x + 20) + (2x + 40) = 180
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (4x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 4x + 20
   
   (2x + 40) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 40
   
   The equation is now 4x + 20 + 2x + 40 = 180
3. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 4x + 2x make 6x and the numbers 20 + 40 make 60.
   The equation is now 6x + 60 = 180
4. Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   6x + 60 − 60 = 180 − 60
   6x = 120
5. Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.
   6x6 = 1206
   x = 1206 = 20
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 20
7. Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.
   x = 20. The angles are 100° and 80°.

Check. 100 + 80 = 180 ✓

Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   5x + 3 = 38
2. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   5x + 3 − 3 = 38 − 3
   5x = 35
3. Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.
   5x5 = 355
   x = 355 = 7
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 7

Check. Substitute x = 7 into the original equation. Left side: 5 × 7 + 3 = 35 + 3 = 38. Right side: 38. Both sides equal 38, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   x4 + 3 = 8
2. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x4 + 3 − 3 = 8 − 3
   x4 = 5
3. Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.
   4 × x4 = 4 × 5
   x = 20
4. Write the solution. x is now on its own, so the equation tells us its value.
   x = 20

Check. Substitute x = 20 into the original equation. Left side: 204 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓

Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   7x − 3 = 3x + 17
2. Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   7x − 3 − 3x = 3x + 17 − 3x
   4x − 3 = 17
3. Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x − 3 + 3 = 17 + 3
   4x = 20
4. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 204
   x = 204 = 5
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 5

Check. Substitute x = 5 into the original equation. Left side: 7 × 5 − 3 = 35 − 3 = 32. Right side: 3 × 5 + 17 = 15 + 17 = 32. Both sides equal 32, so the solution is correct. ✓

Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   3(2x − 1) = 21
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 3(2x − 1):
   3 × 2x = 6x
   3 × (−1) = −3
   
   The equation is now 6x − 3 = 21
3. Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   6x − 3 + 3 = 21 + 3
   6x = 24
4. Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.
   6x6 = 246
   x = 246 = 4
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 4

Check. Substitute x = 4 into the original equation. Left side: 3 × (2 × 4 − 1) = 3 × 7 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓

Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.

1. Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.
   5(x − 2) = 3(x + 4)
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 5(x − 2):
   5 × x = 5x
   5 × (−2) = −10
   
   Expand 3(x + 4):
   3 × x = 3x
   3 × 4 = 12
   
   The equation is now 5x − 10 = 3x + 12
3. Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   5x − 10 − 3x = 3x + 12 − 3x
   2x − 10 = 12
4. Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   2x − 10 + 10 = 12 + 10
   2x = 22
5. Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.
   2x2 = 222
   x = 222 = 11
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 11

Check. Substitute x = 11 into the original equation. Left side: 5 × (11 − 2) = 5 × 9 = 45. Right side: 3 × (11 + 4) = 3 × 15 = 45. Both sides equal 45, so the solution is correct. ✓

Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.

1. Use angles on a straight line. Angles on a straight line add up to 180°.
   (5x + 10) + (3x + 30) + 2x = 180
   Equation: (5x + 10) + (3x + 30) + 2x = 180
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (5x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 5x + 10
   
   (3x + 30) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 30
   
   The equation is now 5x + 10 + 3x + 30 + 2x = 180
3. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 5x + 3x + 2x make 10x and the numbers 10 + 30 make 40.
   The equation is now 10x + 40 = 180
4. Subtract 40 from both sides. This undoes the +40, because +40 − 40 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   10x + 40 − 40 = 180 − 40
   10x = 140
5. Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.
   10x10 = 14010
   x = 14010 = 14
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 14
7. Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.
   x = 14. The angles are 80°, 72° and 28°.

Check. 80 + 72 + 28 = 180 ✓

Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.

Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.

1. Use the perimeter. The perimeter is two lengths plus two widths.
   Perimeter = 2(3x + 1) + 2(2x − 3) = 46
   Equation: 2(3x + 1) + 2(2x − 3) = 46
2. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 2(3x + 1):
   2 × 3x = 6x
   2 × 1 = 2
   
   Expand 2(2x − 3):
   2 × 2x = 4x
   2 × (−3) = −6
   
   The equation is now 6x + 2 + 4x − 6 = 46
3. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms 6x + 4x make 10x and the numbers 2 − 6 make −4.
   The equation is now 10x − 4 = 46
4. Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   10x − 4 + 4 = 46 + 4
   10x = 50
5. Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.
   10x10 = 5010
   x = 5010 = 5
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 5
7. Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.
   x = 5. The sides are 16 cm and 7 cm, so the area is 16 × 7 = 112 cm².

Check. 2 × 16 + 2 × 7 = 46 cm ✓

Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.

1. Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.
   Let x = the cost of one T-shirt in pounds.
2. Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.
   4 T-shirts cost 4x pounds, and delivery of £3 is added once. The total is £35.
   Equation: 4x + 3 = 35
3. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x + 3 − 3 = 35 − 3
   4x = 32
4. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 324
   x = 324 = 8
5. Write the solution. x is now on its own, so the equation tells us its value.
   x = 8
6. Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.
   One T-shirt costs £8.

Check. 4 × £8 = £32, and £32 + £3 = £35 ✓

Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.

1. Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.
   Let x = Ben's age in years.
2. Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.
   Amy is 3 years older than Ben: x + 3. Cal is twice as old as Ben: 2x. The three ages add up to 39.
   Equation: x + (x + 3) + 2x = 39
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   (x + 3) has nothing in front of it except +1, so the brackets can simply be removed: x + 3
   
   The equation is now x + x + 3 + 2x = 39
4. Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.
   On the left side: the x terms x + x + 2x make 4x.
   The equation is now 4x + 3 = 39
5. Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   4x + 3 − 3 = 39 − 3
   4x = 36
6. Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.
   4x4 = 364
   x = 364 = 9
7. Write the solution. x is now on its own, so the equation tells us its value.
   x = 9
8. Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.
   Ben is 9, Amy is 12 and Cal is 18.

Check. 9 + 12 + 18 = 39 ✓

Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Substitute the value into both sides of the equation, work each side out, and compare.

1. Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).
   5x − 7 = 3x + 1,   x = 4
2. Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.
   Left side = 5 × 4 − 7 = 20 − 7 = 13
3. Substitute into the right-hand side. Do exactly the same to the other side of the equation.
   Right side = 3 × 4 + 1 = 12 + 1 = 13
4. Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.
   13 = 13, so yes, x = 4 is a solution.

Check. The two sides match, which is exactly what we wanted to see. ✓

Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.

1. Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.
   Let x = the number of minutes.
2. Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.
   Deal A costs 1000 + 4x pence. Deal B costs 400 + 10x pence. We want the costs to be equal.
   Equation: 4x + 1000 = 10x + 400
3. Remove the x term from the left side: subtract 4x from both sides. We want all the x terms on one side. The left side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   4x + 1000 − 4x = 10x + 400 − 4x
   1000 = 6x + 400
4. Subtract 400 from both sides. This undoes the +400, because +400 − 400 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   1000 − 400 = 6x + 400 − 400
   600 = 6x
5. Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.
   6006 = 6x6
   x = 6006 = 100
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 100
7. Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.
   The deals cost the same for 100 minutes.

Check. Deal A: 1000 + 4 × 100 = 1400p. Deal B: 400 + 10 × 100 = 1400p. The same ✓

Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Substitute the value into both sides of the equation, work each side out, and compare.

1. Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).
   2x + 5 = 4x − 1,   x = 3
2. Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.
   Left side = 2 × 3 + 5 = 6 + 5 = 11
3. Substitute into the right-hand side. Do exactly the same to the other side of the equation.
   Right side = 4 × 3 − 1 = 12 − 1 = 11
4. Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.
   11 = 11, so yes, x = 3 is a solution.

Check. The two sides match, which is exactly what we wanted to see. ✓

Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.

Key words. To solve an equation means to find the value of x that makes it true. Whatever you do to one side you must do to the other side, so that it stays balanced, like a pair of scales.

Method. Solve an equation containing fractions: (1) find the common denominator, (2) multiply every term by it to remove the fractions, (3) solve the simple equation that is left.

1. Write the equation. An equation says two things are equal. We want the value of x that makes both sides equal.
   4x − 35 − 2x + 24 = 1
2. Find the common denominator of all the bottoms. Fractions are awkward to solve with, so we will remove them. To do that we multiply by something that every bottom divides into: the lowest common multiple of the bottoms.
   5, 4
   Common denominator = 20
3. Multiply every term, on both sides, by the common denominator. Doing the same thing to both sides keeps the equation balanced. Multiply every single term, including any plain numbers. This removes all the fractions.
   20 × 4x − 35 = 4(4x − 3)   (the bottom 5 goes into 20 exactly 4 times, so the bottom disappears and the top is multiplied by 4)
   20 × 2x + 24 = 5(2x + 2)   (the bottom 4 goes into 20 exactly 5 times, so the bottom disappears and the top is multiplied by 5)
   20 × 1 = 20   (a plain number or term with no bottom is simply multiplied by 20)
4. Write the new equation. There are no fractions left. Check that every term from the original equation has been multiplied.
   4(4x − 3) − 5(2x + 2) = 20
5. Expand the brackets. Multiply each term inside by the number or letter outside. A minus sign in front of a bracket changes the sign of every term inside it.
   Expand 4(4x − 3):
   4 × 4x = 16x
   4 × (−3) = −12
   so it is 16x − 12
   
   Expand 5(2x + 2):
   5 × 2x = 10x
   5 × 2 = 10
   so it is 10x + 10
6. Collect like terms on each side. Add or subtract the x terms together, and the plain numbers together, on each side separately.
   16x − 12 − 10x − 10 = 20
   6x − 22 = 20
7. Get the x terms on one side and the plain numbers on the other. Use inverse operations on both sides. The inverse of adding is subtracting and the inverse of subtracting is adding. Always do the same to both sides.
   Start: 6x − 22 = 20
   Now get the numbers on the other side. Add 22 to both sides: 6x = 42
8. Divide both sides by the number in front of x (which is 6). 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6. This leaves x on its own.
   x = 42 ÷ 6 = 7

Check. Put x = 7 back into the original equation. Left side: 25/5 − 16/4 = 1. Right side: 1 = 1. Both sides are equal, so the solution is correct. ✓

Watch out. Multiply every term, including the numbers that are not fractions. And a solution is only allowed if it does not make a bottom equal to 0.

Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.

Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.

1. Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.
   Let x = Eli's age now, in years.
2. Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.
   Dan is 4 times as old as Eli: 4x. In 6 years Eli will be x + 6 and Dan will be 4x + 6. Then Dan is 3 times as old as Eli: 4x + 6 = 3(x + 6).
   Equation: 4x + 6 = 3(x + 6)
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 3(x + 6):
   3 × x = 3x
   3 × 6 = 18
   
   The equation is now 4x + 6 = 3x + 18
4. Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.
   4x + 6 − 3x = 3x + 18 − 3x
   x + 6 = 18
5. Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   x + 6 − 6 = 18 − 6
   x = 12
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 12
7. Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.
   Eli is 12 and Dan is 48.

Check. In 6 years: Eli 18, Dan 54, and 54 = 3 × 18 ✓

Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.

Key words. The denominator is the bottom of a fraction. We cannot divide by zero, so a value of x that makes a denominator zero is never a solution.

Method. Multiply both sides by the bottom of the fraction to remove it, then solve the ordinary equation that is left.

1. Write down the equation. The unknown is in the bottom (denominator) of a fraction. The bottom can never be zero, so we will check this at the end.
   12x − 1 = 3
2. Multiply both sides by (x − 1). This removes the fraction. Multiplying by the bottom cancels it, because anything divided by itself is 1. We do it to both sides to keep the balance.
   12x − 1 × (x − 1) = 3 × (x − 1)
   12 = 3(x − 1)
3. Expand the brackets. Multiply each term inside the bracket by the number outside it.
   Expand 3(x − 1):
   3 × x = 3x
   3 × (−1) = −3
   
   The equation is now 12 = 3x − 3
4. Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.
   12 + 3 = 3x − 3 + 3
   15 = 3x
5. Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.
   153 = 3x3
   x = 153 = 5
6. Write the solution. x is now on its own, so the equation tells us its value.
   x = 5
7. Check that the bottom is not zero. If a solution made the bottom equal to 0 the fraction would be impossible, so it would have to be rejected.
   x − 1 = 4 when x = 5, which is not 0, so the solution is allowed.

Check. Substitute x = 5: the bottom is 4, so the fraction is 124 = 3, which equals 3. ✓

Watch out. Multiply the whole of the other side by the bottom, so the right-hand side becomes a bracket. Never solve by "flipping" only part of it.

Method. Expand and simplify each side, then see what is left.

1. Expand the brackets. Multiply every term inside by the number outside.
   4(x + 2) = 4x + 8   and   2(x + 5) = 2x + 10
2. Collect like terms on the left. 4x − 2x = 2x.
   2x + 8 = 2x + 10
3. Subtract 2x from both sides. The x terms cancel.
   8 = 10
4. Conclude. 8 = 10 is never true, whatever x is.
   The equation has no solution.

Check. Try x = 3: left side 4 × 5 − 6 = 14, right side 2 × 8 = 16. The two sides always differ by 2. ✓

Watch out. Marks: 1 for expanding correctly, 1 for simplifying to 2x + 8 = 2x + 10, 1 for the conclusion.