Divide by the factor

The idea

If one side of an equation is a product of two or more numbers or expressions and one factor is removed, the other side must be divided by that factor for the equality to hold.

2y = 14

If y = 7, then 2y = 14. Both sides are 14. Now we remove the factor 2 from the left side. The left side becomes just y, which is 7.

What must the right side, 14, become so that the sides still agree?

• 14 ÷ 2
• 14 × 2
• 14 − 2

Remove the factor, then repair

Tap the highlighted factor to remove it. Then choose what the other side must become and test it with the numbers.

Multiplication is undone by division

2y means 2 times y. Taking away the factor 2 leaves *y*, which is 2 times smaller. So the other side must also be made 2 times smaller: divided by 2.

This is the same as dividing both sides by 2: 2y ÷ 2 = 14 ÷ 2.

If one side of an equation is a product and one factor is removed, the other side must be divided by that factor: 2y = 14 becomes y = 14 ÷ 2.

• Equation | After removing the factor
• 2y = 14 | y = 14 ÷ 2
• 11y = 66 | y = 66 ÷ 11
• 36 = 4m | 36 ÷ 4 = m
• 3x = −12 | x = −12 ÷ 3

Notes

A factor removed from a product on one side means the other side is divided by that factor: 2y = 14 becomes y = 14 ÷ 2.

Check yourself

In 5w = 60, the factor 5 is removed from the left. What does the right side become?

Solve 7k = 91.

Answer: 13

k = 91 ÷ 7 = 13. Check: 7 × 13 = 91.

Solve −4p = 20.

Answer: -5

p = 20 ÷ (−4) = −5. Check: −4 × (−5) = 20.

To solve 3x + 1 = 22, a student divides only the 3x by 3 and writes *x* + 1 = 22 ÷ 3. What is wrong?

• 60 × 5. That makes the right side bigger. Removing a factor makes the left side 5 times smaller, so the right side must be divided.
• 60 ÷ 5 — correct. Yes! Multiplication is undone by division.
• 60 − 5. A factor is removed, so the right side is divided by it, not reduced by it.
• Nothing: dividing 3x by 3 gives x, and 22 by 3 gives 22 ÷ 3.. The 1 would also have to be divided by 3, which it was not. Remove the +1 first, then the factor.
• The factor 3 should be added to the right side instead.. A factor is removed by dividing, not by adding.
• The 3 is a factor of 3x only. First remove the +1, then divide the whole of the other side: 3x = 21. — correct. Yes! 3x = 22 − 1 = 21 and then x = 21 ÷ 3 = 7.