Solving by multiplying or dividing

If 4 identical notebooks cost 12 dollars, one costs 12 ÷ 4 = 3. That is the equation 4n = 12, solved by dividing both sides by 4.

When the variable is multiplied

In the last unit the variable had a number added to it or taken away. Here the variable is multiplied by a number. In 4n = 12, n has been multiplied by 4. Subtracting 4 would not help, because there is no 4 standing on its own to remove. The opposite of multiplying by 4 is dividing by 4, so divide both sides by 4: 4n ÷ 4 = 12 ÷ 4, which leaves n = 3. This is the Division Property of Equality: if a = b and c is not zero, then a ÷ c = b ÷ c. On the balance, it means splitting both pans into four equal groups and keeping one group from each side.

Why must c not be zero? Division by zero has no meaning, so the property only works with a non-zero divisor. The same caution applies to the next property. The Multiplication Property of Equality says that if a = b, then a × c = b × c. It undoes division. In x ÷ 5 = 6, which is also written x/5 = 6, multiply both sides by 5 to get x = 30. Check it: 30 ÷ 5 = 6. Multiplying both sides by zero would make any equation true, 0 = 0, and tell you nothing, so we never use zero here either.

Negative and fractional coefficients

Signs need care. In −7y = 42, the coefficient is −7, so divide both sides by −7: y = 42 ÷ (−7) = −6. A quick check: −7 × (−6) = 42. A special case is −x = 5. The coefficient there is −1, even though no 1 is written. Divide both sides by −1, or multiply by −1, and you get x = −5. Many slips come from dividing by 7 instead of −7, which flips the sign of the answer. Reading the coefficient with its sign before you divide prevents that.

When the coefficient is a fraction, multiplying by its reciprocal is quicker than dividing. In (2/3)x = 18, the reciprocal of 2/3 is 3/2, and 3/2 × 2/3 = 1. Multiply both sides by 3/2: x = 18 × 3/2 = 27. Check: two thirds of 27 is 18. You could also think of it in two steps, multiplying by 3 to get 2x = 54 and then dividing by 2, and you land on the same 27. The reciprocal does both moves at once.

In the graph lab below the blue line is y = 3x, the left side of 3x = 12, and the orange line is y = 12. They meet above x = 4. Make the coefficient a negative number and the blue line tilts the other way, so the crossing point moves to the left of zero. Set a to 0 and the blue line lies flat; unless d is also 0, it never meets the orange line, which matches the rule that you cannot divide by zero.