Take the same from both pans

The idea

When unknown terms appear on both sides of an equation, subtracting the same unknown term from both sides brings them to one side, after which the equation is solved as before.

Sacks on both pans

A scale balances 6 sacks and a weight of 7 against 4 sacks and a weight of 21. All sacks weigh the same, *y*. The equation is 6y + 7 = 4y + 21.

What could we take off both pans to make the problem simpler?

• 4 sacks
• a weight of 21
• We cannot take anything

Empty one pan of sacks

Take the same from both pans, or share both pans into equal parts. The equation under the scale changes with every move. Stop when one sack is alone.

Bring the unknowns to one side

Taking 4 sacks off each pan keeps the scale balanced. In symbols, we subtract 4y from both sides. Then the unknown is only on one side, and we solve as before.

Check: 6 × 7 + 7 = 49 and 4 × 7 + 21 = 49.

When unknown terms are on both sides, subtract the same unknown term from both sides to bring them to one side. After that, solve the equation as before.

• Step | Equation
• start | 6y + 7 = 4y + 21
• subtract 4y from both sides | 6y + 7 − 4y = 4y + 21 − 4y, so 2y + 7 = 21
• subtract 7 from both sides | 2y = 14
• divide both sides by 2 | *y* = 7

Notes

With unknown terms on both sides, subtract the same unknown term from both sides to bring them to one side: 6y + 7 = 4y + 21 → 2y + 7 = 21 → *y* = 7.

Check yourself

In 8x + 7 = 3x + 32, which move brings the *x* terms to one side?

Solve 8x + 7 = 3x + 32.

Answer: 5

8x + 7 − 3x = 32 gives 5x + 7 = 32, then 5x = 25, so x = 5. Check: 8 × 5 + 7 = 47 and 3 × 5 + 32 = 47.

Aman subtracts 4y from the left side only: 6y + 7 − 4y = 4y + 21. What is wrong?

Solve 7k + 1 = 2k − 14.

Answer: -3

7k + 1 − 2k = −14 gives 5k + 1 = −14, then 5k = −15, so k = −3. Check: 7 × (−3) + 1 = −20 and 2 × (−3) − 14 = −20.

• Subtract 3x from both sides. — correct. Yes! It leaves 5x + 7 = 32, with *x* on one side only.
• Subtract 3x from the right side only.. The two sides would no longer be equal. The same must be taken from both sides.
• Subtract 7 from both sides.. That removes the loose 7, but there are still *x* terms on both sides: 8x = 3x + 25.
• He should have added 4y instead.. Adding 4y would put even more sacks on the pans. Taking 4y off both sides is what removes the sacks.
• Nothing: subtracting from one side is allowed.. Not so. Taking something off one pan only makes the scale tip.
• The same 4y must also be subtracted from the right side. — correct. Yes! It must be 4y + 21 − 4y. Then both sides stay equal.