Do the same to both sides

The idea

Since both sides of an equation have the same value, adding, subtracting, multiplying or dividing both sides by the same number keeps them equal, just as removing equal weights from both plates keeps a scale balanced.

A balanced scale

A scale balances 15 + 8 on the left with 23 on the right. Both pans weigh the same.

You add a weight of 10 to the left pan only. What happens?

• The scale stays balanced
• The left pan goes down
• The right pan goes down

Treat both sides alike

Pick an operation, a number and a side. Do it to both sides, then try it on one side only. Watch the scale and the equal sign.

Both sides, never one

The two sides of an equation have the same value. If you treat both sides in the same way, they still have the same value. If you treat only one side, they do not.

It is like taking the same weight off both plates of a balanced scale: it stays balanced.

Both sides of an equation have the same value. So adding, subtracting, multiplying or dividing both sides by the same number keeps them equal, just as removing equal weights from both plates keeps a scale balanced.

• Do to both sides | Result
• start | 15 + 8 = 23
• add 10 | 15 + 8 + 10 = 23 + 10, so 33 = 33
• subtract 5 | 15 + 8 − 5 = 23 − 5, so 18 = 18
• multiply by 2 | (15 + 8) × 2 = 23 × 2, so 46 = 46
• divide by 4 (12 + 8 = 20) | (12 + 8) ÷ 4 = 20 ÷ 4, so 5 = 5

Notes

Both sides of an equation have the same value, so the same operation on both sides (add, subtract, multiply or divide by the same number) keeps them equal.

Check yourself

We start with 15 + 8 = 23. Subtract 5 from both sides. Which statement do we get?

Start with 15 + 8 = 23. Add 7 to both sides. What do both sides become?

Answer: 30

15 + 8 + 7 = 30 and 23 + 7 = 30. Both sides are still equal.

Which move keeps 14 + 6 = 20 true?

Multiply both sides of 9 + 3 = 12 by 2. What does the LHS, (9 + 3) × 2, become?

Answer: 24

(9 + 3) × 2 = 12 × 2 = 24, and the RHS is 12 × 2 = 24 as well.

• 15 + 8 − 5 = 23. The 5 was taken from the LHS only. Then 18 ≠ 23.
• 15 + 8 − 5 = 23 − 5 — correct. Yes! The 5 is taken from the LHS and from the RHS.
• 15 + 8 − 5 = 23 + 5. Adding 5 on the right is not the same operation. Then 18 ≠ 28.
• Subtract 4 from the left side only.. Then the LHS is 16 but the RHS is still 20. They are no longer equal.
• Subtract 4 from the left side and add 4 to the right side.. These are different operations. The LHS becomes 16 and the RHS becomes 24.
• Subtract 4 from both sides. — correct. Yes! 14 + 6 − 4 = 16 and 20 − 4 = 16.