Direct proportion equations

Objective. Use y = kx to find and apply a constant of proportionality.

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Learn: the key idea

Direct proportion means two quantities increase or decrease together at a constant rate, described by the equation y = kx, where k is the constant of proportionality. To find k, substitute one known pair of x and y values into the equation and solve for k, then use that value to predict any other pair. For example, if y is directly proportional to x and y = 20 when x = 4, then k = 20 ÷ 4 = 5, giving the equation y = 5x, which can then find y when x = 7 as 35. Graphically, direct proportion always produces a straight line through the origin.

Worked example

y is directly proportional to x. When x = 6, y = 18. Find the value of y when x = 10.

1 $Use y = kx and substitute the known values: 18 = k \times 6.$
2 $Divide both sides by 6 to find k: k = 3.$
3 $Write the equation: y = 3x.$
4 $Substitute x = 10: y = 3 \times 10 = 30.$

Answer: y = 30

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Common mistake. Pupils sometimes find k by subtracting x from y instead of dividing, treating the relationship as additive rather than multiplicative. Always divide y by x to find the constant of proportionality k.

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