GCSE Maths: The unknown on both sides (Higher)
Part of the GCSE Maths linear equations chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
The unknown on both sides
Often the letter appears on both sides of the equals sign, such as 3x + 2 = x + 8. Now we cannot get x on its own just by undoing one side. First we must collect all the x terms on one side.
Think of the scales again. The left pan has three boxes and two circles, the right pan has one box and eight circles. Take one box off both pans and the scales still balance.
Here is the method.
- Choose the smaller x term (the one with the smaller number in front of x) and remove it from both sides. Subtracting the smaller one means the x terms that are left are positive.
- Move the plain numbers to the other side: add or subtract the number that is on the same side as the x term, on both sides.
- Divide both sides by the number in front of x.
For 3x + 2 = x + 8, the smaller x term is x (that is 1x). Subtract x from both sides to get 2x + 2 = 8. Then subtract 2 to get 2x = 6, and divide by 2 to get x = 3.
What if the bigger x term is on the right? In 3x + 10 = 7x − 6, the smaller x term is 3x on the left. Subtract 3x from both sides: 10 = 4x − 6. Add 6: 16 = 4x. Divide by 4: 4 = x, which means x = 4. You could remove the bigger x term instead, but then you get a negative number of x and mistakes with signs become more likely. Both ways give the same answer.
Key ideas
- Remove the smaller x term from both sides, so the x terms combine on one side.
- Then move the plain numbers to the other side, and divide.
- When you subtract a term from both sides, subtract it from every part of both sides, and watch the signs.
- Check by substituting into both sides of the original equation.
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.
- 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 = 2x + 12
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x 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 + 3 − 2x = 2x + 12 − 2x
3x + 3 = 12 - 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.3x + 3 − 3 = 12 − 3
3x = 9 - 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 = 93
x = 93 = 3 - 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: 5 × 3 + 3 = 15 + 3 = 18. Right side: 2 × 3 + 12 = 6 + 12 = 18. Both sides equal 18, 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.
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.
- 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 − 4 = 3x + 20
- 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 − 4 − 3x = 3x + 20 − 3x
4x − 4 = 20 - 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.4x − 4 + 4 = 20 + 4
4x = 24 - 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 = 244
x = 244 = 6 - 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: 7 × 6 − 4 = 42 − 4 = 38. Right side: 3 × 6 + 20 = 18 + 20 = 38. Both sides equal 38, 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.
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.
- 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 + 10 = 7x − 6
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left 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.3x + 10 − 3x = 7x − 6 − 3x
10 = 4x − 6 - Add 6 to 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.10 + 6 = 4x − 6 + 6
16 = 4x - 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.164 = 4x4
x = 164 = 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: 3 × 4 + 10 = 12 + 10 = 22. Right side: 7 × 4 − 6 = 28 − 6 = 22. Both sides equal 22, 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.
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.
- 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 = 5x + 15
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x + 9 − 2x = 5x + 15 − 2x
9 = 3x + 15 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.9 − 15 = 3x + 15 − 15
−6 = 3x - 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.−63 = 3x3
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 2 × (−2) + 9 = −4 + 9 = 5. Right side: 5 × (−2) + 15 = −10 + 15 = 5. Both sides equal 5, 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.
Watch out
Take the x term away from both sides, and think about the sign. In 5x − 3 = 2x + 9, subtracting 2x from both sides gives 3x − 3 = 9. If you add 2x to the left (a common mistake) you get 7x, which is wrong.Exercise 9D The unknown on both sides
Non-calculatorKey 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.
- 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 = 3x + 8
- 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 − 3x = 3x + 8 − 3x
2x = 8 - 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 = 82
x = 82 = 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 = 20. Right side: 3 × 4 + 8 = 12 + 8 = 20. Both sides equal 20, 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.
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.
- 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 = x + 12
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x 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 − x = x + 12 − x
3x = 12 - 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 = 123
x = 123 = 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: 4 × 4 = 16. Right side: 4 + 12 = 16. Both sides equal 16, 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.
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.
- 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 = 4x + 15
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right 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.7x − 4x = 4x + 15 − 4x
3x = 15 - 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 = 153
x = 153 = 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 = 35. Right side: 4 × 5 + 15 = 20 + 15 = 35. Both sides equal 35, 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.
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.
- 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.6x = 2x + 20
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x 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.6x − 2x = 2x + 20 − 2x
4x = 20 - 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 - 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: 6 × 5 = 30. Right side: 2 × 5 + 20 = 10 + 20 = 30. Both sides equal 30, 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.
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.
- 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.9x = 5x + 28
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x 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.9x − 5x = 5x + 28 − 5x
4x = 28 - 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 = 284
x = 284 = 7 - 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: 9 × 7 = 63. Right side: 5 × 7 + 28 = 35 + 28 = 63. Both sides equal 63, 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.
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.
- 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.8x = 3x + 35
- 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.8x − 3x = 3x + 35 − 3x
5x = 35 - 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 - 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: 8 × 7 = 56. Right side: 3 × 7 + 35 = 21 + 35 = 56. Both sides equal 56, 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.
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.
- 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 + 5 = x + 11
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x 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.3x + 5 − x = x + 11 − x
2x + 5 = 11 - 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.2x + 5 − 5 = 11 − 5
2x = 6 - 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 - 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: 3 × 3 + 5 = 9 + 5 = 14. Right side: 3 + 11 = 14. Both sides equal 14, 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.
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.
- 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 + 2 = 2x + 14
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x 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 + 2 − 2x = 2x + 14 − 2x
3x + 2 = 14 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 2 − 2 = 14 − 2
3x = 12 - 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 = 123
x = 123 = 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 + 2 = 20 + 2 = 22. Right side: 2 × 4 + 14 = 8 + 14 = 22. Both sides equal 22, 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.
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.
- 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 + 1 = x + 16
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x 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 + 1 − x = x + 16 − x
3x + 1 = 16 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 16 − 1
3x = 15 - 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 = 153
x = 153 = 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: 4 × 5 + 1 = 20 + 1 = 21. Right side: 5 + 16 = 21. Both sides equal 21, 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.
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.
- 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.6x + 3 = 4x + 11
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right 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.6x + 3 − 4x = 4x + 11 − 4x
2x + 3 = 11 - 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.2x + 3 − 3 = 11 − 3
2x = 8 - 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 = 82
x = 82 = 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: 6 × 4 + 3 = 24 + 3 = 27. Right side: 4 × 4 + 11 = 16 + 11 = 27. Both sides equal 27, 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.
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.
- 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 + 4 = 3x + 20
- 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 + 4 − 3x = 3x + 20 − 3x
4x + 4 = 20 - Subtract 4 from 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.4x + 4 − 4 = 20 − 4
4x = 16 - 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 - 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 + 4 = 28 + 4 = 32. Right side: 3 × 4 + 20 = 12 + 20 = 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.
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.
- 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.8x + 1 = 5x + 19
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x 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.8x + 1 − 5x = 5x + 19 − 5x
3x + 1 = 19 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 19 − 1
3x = 18 - 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 - 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: 8 × 6 + 1 = 48 + 1 = 49. Right side: 5 × 6 + 19 = 30 + 19 = 49. Both sides equal 49, 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.
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.
- 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 = 2x + 9
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x 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 − 3 − 2x = 2x + 9 − 2x
3x − 3 = 9 - 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.3x − 3 + 3 = 9 + 3
3x = 12 - 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 = 123
x = 123 = 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 − 3 = 20 − 3 = 17. Right side: 2 × 4 + 9 = 8 + 9 = 17. Both sides equal 17, 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.
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.
- 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.6x − 4 = 3x + 14
- 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.6x − 4 − 3x = 3x + 14 − 3x
3x − 4 = 14 - 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 = 14 + 4
3x = 18 - 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 - 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: 6 × 6 − 4 = 36 − 4 = 32. Right side: 3 × 6 + 14 = 18 + 14 = 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.
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.
- 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 − 7 = x + 8
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x 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 − 7 − x = x + 8 − x
3x − 7 = 8 - Add 7 to 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 = 8 + 7
3x = 15 - 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 = 153
x = 153 = 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: 4 × 5 − 7 = 20 − 7 = 13. Right side: 5 + 8 = 13. Both sides equal 13, 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.
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.
- 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 − 1 = 5x + 13
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x 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 − 1 − 5x = 5x + 13 − 5x
2x − 1 = 13 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 1 + 1 = 13 + 1
2x = 14 - 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 = 142
x = 142 = 7 - 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 × 7 − 1 = 49 − 1 = 48. Right side: 5 × 7 + 13 = 35 + 13 = 48. Both sides equal 48, 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.
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.
- 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.9x − 8 = 4x + 27
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right 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.9x − 8 − 4x = 4x + 27 − 4x
5x − 8 = 27 - 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 = 27 + 8
5x = 35 - 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 - 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: 9 × 7 − 8 = 63 − 8 = 55. Right side: 4 × 7 + 27 = 28 + 27 = 55. Both sides equal 55, 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.
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.
- 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.8x − 5 = 3x + 20
- 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.8x − 5 − 3x = 3x + 20 − 3x
5x − 5 = 20 - 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.5x − 5 + 5 = 20 + 5
5x = 25 - 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 = 255
x = 255 = 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: 8 × 5 − 5 = 40 − 5 = 35. Right side: 3 × 5 + 20 = 15 + 20 = 35. Both sides equal 35, 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.
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.
- 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 + 12 = 3x + 4
- Remove the x term from the left side: subtract x from both sides. We want all the x terms on one side. The left side has x; subtracting x 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.x + 12 − x = 3x + 4 − x
12 = 2x + 4 - Subtract 4 from 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.12 − 4 = 2x + 4 − 4
8 = 2x - 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.82 = 2x2
x = 82 = 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: 4 + 12 = 16. Right side: 3 × 4 + 4 = 12 + 4 = 16. Both sides equal 16, 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.
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.
- 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 = 5x
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x + 9 − 2x = 5x − 2x
9 = 3x - 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.93 = 3x3
x = 93 = 3 - 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 = 15. Right side: 5 × 3 = 15. Both sides equal 15, 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.
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.
- 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 + 8 = 5x − 4
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left 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.3x + 8 − 3x = 5x − 4 − 3x
8 = 2x − 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.8 + 4 = 2x − 4 + 4
12 = 2x - 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.122 = 2x2
x = 122 = 6 - 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 + 8 = 18 + 8 = 26. Right side: 5 × 6 − 4 = 30 − 4 = 26. Both sides equal 26, 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.
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.
- 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 + 3 = 7x − 18
- 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 + 3 − 4x = 7x − 18 − 4x
3 = 3x − 18 - Add 18 to both sides. This undoes the −18, because −18 + 18 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3 + 18 = 3x − 18 + 18
21 = 3x - 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.213 = 3x3
x = 213 = 7 - 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: 4 × 7 + 3 = 28 + 3 = 31. Right side: 7 × 7 − 18 = 49 − 18 = 31. Both sides equal 31, 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.
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.
- 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 + 15 = 6x − 13
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x + 15 − 2x = 6x − 13 − 2x
15 = 4x − 13 - Add 13 to both sides. This undoes the −13, because −13 + 13 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.15 + 13 = 4x − 13 + 13
28 = 4x - 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.284 = 4x4
x = 284 = 7 - 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: 2 × 7 + 15 = 14 + 15 = 29. Right side: 6 × 7 − 13 = 42 − 13 = 29. Both sides equal 29, 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.
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.
- 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 = 8x − 23
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left 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.3x + 7 − 3x = 8x − 23 − 3x
7 = 5x − 23 - Add 23 to both sides. This undoes the −23, because −23 + 23 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.7 + 23 = 5x − 23 + 23
30 = 5x - 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.305 = 5x5
x = 305 = 6 - 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: 8 × 6 − 23 = 48 − 23 = 25. Both sides equal 25, 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.
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.
- 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 = 5x + 15
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x + 9 − 2x = 5x + 15 − 2x
9 = 3x + 15 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.9 − 15 = 3x + 15 − 15
−6 = 3x - 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.−63 = 3x3
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 2 × (−2) + 9 = −4 + 9 = 5. Right side: 5 × (−2) + 15 = −10 + 15 = 5. Both sides equal 5, 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.
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.
- 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 + 14 = 7x + 22
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left 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.3x + 14 − 3x = 7x + 22 − 3x
14 = 4x + 22 - Subtract 22 from both sides. This undoes the +22, because +22 − 22 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.14 − 22 = 4x + 22 − 22
−8 = 4x - 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.−84 = 4x4
x = −84 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 3 × (−2) + 14 = −6 + 14 = 8. Right side: 7 × (−2) + 22 = −14 + 22 = 8. Both sides equal 8, 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.
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.
- 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 + 4 = 8x + 13
- Remove the x term from the left side: subtract 5x from both sides. We want all the x terms on one side. The left side has 5x; subtracting 5x 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 + 4 − 5x = 8x + 13 − 5x
4 = 3x + 13 - Subtract 13 from both sides. This undoes the +13, because +13 − 13 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4 − 13 = 3x + 13 − 13
−9 = 3x - 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.−93 = 3x3
x = −93 = −3 - 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: 5 × (−3) + 4 = −15 + 4 = −11. Right side: 8 × (−3) + 13 = −24 + 13 = −11. Both sides equal −11, 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.
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.
- 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 − 3 = 4x + 7
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x − 3 − 2x = 4x + 7 − 2x
−3 = 2x + 7 - 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.−3 − 7 = 2x + 7 − 7
−10 = 2x - 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.−102 = 2x2
x = −102 = −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: 2 × (−5) − 3 = −10 − 3 = −13. Right side: 4 × (−5) + 7 = −20 + 7 = −13. Both sides equal −13, 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.
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.
- 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 + 1 = 4x + 10
- Remove the x term from the left side: subtract x from both sides. We want all the x terms on one side. The left side has x; subtracting x 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.x + 1 − x = 4x + 10 − x
1 = 3x + 10 - Subtract 10 from 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.1 − 10 = 3x + 10 − 10
−9 = 3x - 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.−93 = 3x3
x = −93 = −3 - 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: (−3) + 1 = −3 + 1 = −2. Right side: 4 × (−3) + 10 = −12 + 10 = −2. Both sides equal −2, 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.
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.
- 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.6x + 5 = 9x + 20
- Remove the x term from the left side: subtract 6x from both sides. We want all the x terms on one side. The left side has 6x; subtracting 6x 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.6x + 5 − 6x = 9x + 20 − 6x
5 = 3x + 20 - Subtract 20 from both sides. This undoes the +20, because +20 − 20 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5 − 20 = 3x + 20 − 20
−15 = 3x - 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 - 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: 6 × (−5) + 5 = −30 + 5 = −25. Right side: 9 × (−5) + 20 = −45 + 20 = −25. Both sides equal −25, 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.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- 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 miles.
- 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.Firm A charges £3 plus £2 for each mile: 3 + 2x. Firm B charges £1 plus £3 for each mile: 1 + 3x. We want the journey where the prices are equal.
Equation: 3 + 2x = 1 + 3x - 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: write the x term first (adding can be done in any order).
On the right side: write the x term first (adding can be done in any order).
The equation is now 2x + 3 = 3x + 1 - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x 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.2x + 3 − 2x = 3x + 1 − 2x
3 = x + 1 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3 − 1 = x + 1 − 1
2 = x - Write the solution. x is now on its own, so the equation tells us its value.x = 2
- 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 two firms charge the same for a journey of 2 miles.
Check. Firm A: 3 + 2 × 2 = £7. Firm B: 1 + 3 × 2 = £7. 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. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- 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.
- 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.Sam's number multiplied by 5 then 6 subtracted: 5x − 6. The same number multiplied by 3 then 10 added: 3x + 10. The two results are the same.
Equation: 5x − 6 = 3x + 10 - 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 − 6 − 3x = 3x + 10 − 3x
2x − 6 = 10 - Add 6 to 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.2x − 6 + 6 = 10 + 6
2x = 16 - 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 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 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.The number is 8.
Check. 5 × 8 − 6 = 34 and 3 × 8 + 10 = 34 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Find what Lily did wrong, then do the step correctly.
- Look at her first step. The 2x is on the right-hand side. To remove it we subtract 2x from both sides. Lily added 2x to the left-hand side only (6x + 2x = 8x).Wrong: 6x + 2x + 5 = 17
She changed one side only, so the equation is no longer balanced. - Do the step correctly. Subtract 2x from both sides.6x + 5 − 2x = 2x + 17 − 2x
4x + 5 = 17 - Finish solving. Subtract 5, then divide by 4.4x = 12
x = 3
Check. 6 × 3 + 5 = 23 and 2 × 3 + 17 = 23. ✓
Watch out. The same operation must be applied to the whole of both sides.
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.
- 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).7x − 9 = 3x + 11, x = 5
- 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 = 7 × 5 − 9 = 35 − 9 = 26
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3 × 5 + 11 = 15 + 11 = 26
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.26 = 26, so yes, x = 5 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.
- 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 weeks.
- 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.After x weeks Ali has 5 + 4x pounds and Ben has 21 + 2x pounds. We want them to be equal.
Equation: 5 + 4x = 21 + 2x - 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: write the x term first (adding can be done in any order).
On the right side: write the x term first (adding can be done in any order).
The equation is now 4x + 5 = 2x + 21 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x 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 + 5 − 2x = 2x + 21 − 2x
2x + 5 = 21 - 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.2x + 5 − 5 = 21 − 5
2x = 16 - 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 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 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.They have the same amount after 8 weeks.
Check. Ali: 5 + 4 × 8 = £37. Ben: 21 + 2 × 8 = £37. 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.
Method. Finish each method and compare the answers.
- Finish Amy's method. Subtract 8 from both sides, then divide by −4.−4x = −4 − 8 = −12
x = −12 ÷ (−4) = 3 - Finish Ben's method. Add 4 to both sides, then divide by 4.12 = 4x
x = 12 ÷ 4 = 3 - Compare. Both methods give the same value.Both get x = 3. Ben's way avoids dividing by a negative number.
Check. 3 × 3 + 8 = 17 and 7 × 3 − 4 = 17. ✓
Watch out. Moving the smaller x term keeps the number in front of x positive.
Method. Put the known value of x into the equation, then solve for a.
- Substitute x = 3. The solution makes both sides equal, so we can use it.4 × 3 + a = 2 × 3 + 10
12 + a = 16 - Solve for a. Subtract 12 from both sides.a = 16 − 12 = 4
Check. 4 × 3 + 4 = 16 and 2 × 3 + 10 = 16. ✓
Watch out. When a solution is given, substitute it: that turns the equation into one with a single unknown.
Method. Substitute x = 2 and solve the equation that is left for k.
- Substitute x = 2. Both sides must be equal when x is 2.5 × 2 + 3 = k × 2 + 11
13 = 2k + 11 - Solve for k. Subtract 11, then divide by 2.2 = 2k
k = 1
Check. 5 × 2 + 3 = 13 and 1 × 2 + 11 = 13. ✓
Watch out. k is a number in front of x, so kx means k × x.