Same line, new clothes

A messy equation hides its coefficients, the numbers a, b and c. In the lab, slide the number d under y in line B. Line A is y/2 = x/3 + 1. Tidy line B, then see where it meets line A.

The rule

Clear the denominators, the bottom numbers of fractions. Open the brackets by multiplying each term inside. Gather every term on one side, so the other side is 0. Multiply every term by the same number that is not 0. The line does not change. Tidy both equations of a pair before you compare them.

Example

Take y/2 = x/3 + 1. Use 6, because 2 and 3 both divide 6. Multiply every term by 6: 3y = 2x + 6. Here 6 × y/2 = 3y and 6 × x/3 = 2x. That is 2x − 3y + 6 = 0.

Take x/4 + y/6 = 2. Use 12, because 4 and 6 both divide 12. Multiply every term by 12: 3x + 2y = 24. That is 3x + 2y − 24 = 0.

Open a bracket by multiplying each term inside. 2(x − 1) = 3y becomes 2x − 2 = 3y. Here 2 × x = 2x and 2 × 1 = 2. That is 2x − 3y − 2 = 0.