Direct Variation

Consider the amount required to buy a certain number of cinema tickets. No. of tickets bought (x) 1 2 3 4 5 … Amount of money required ($y) 80 160 240 320 400 How does y change when x increases?

Consider the amount required to buy a certain number of cinema tickets. No. of tickets bought (x) 1 2 3 4 5 … Amount of money required ($y) 80 160 240 320 400 When x increases, y increases.

Consider the amount required to buy a certain number of cinema tickets. No. of tickets bought (x) 1 2 3 4 5 … Amount of money required ($y) 80 160 240 320 400 Do you have any further observations about the relation between x and y?

It is observed that the value of is a constant. x y No. of tickets bought (x) 1 2 3 4 5 … Amount of money required ($y) 80 160 240 320 400 x y 1 80 = 80 2 160 = 80 3 240 = 80 4 320 = 80 5 400 = 80 … It is observed that the value of is a constant. x y That is, (or y = 80x). 80 = x y We say that the amount of money required ($y) varies directly as the number of tickets bought (x). Such a relation is called a direct variation (or direct proportion).

Direct Variation In general, if y varies directly as x, x y = k  ‘y varies directly as x’ can also be written as ‘y is directly proportional to x’. x y = k then or y = kx, where k is a non-zero constant. k is called the variation constant. In symbols, we write x. y µ

Consider any two quantities x and y which are in direct variation. i.e. y = kx, where ¹ k The graph of y against x is a straight line (i) passing through the origin (ii) with its slope equal to the variation constant k. y x y = kx Slope = k O

Follow-up question If P µ w, and P = 12 when w = 2, find an equation connecting P and w, the value of w when P = 78. (a) ∵ P µ w k is the variation constant. ∴ P = kw , where k ¹ 0 By substituting P = 12 and w = 2 into the equation, we have 12 = k(2) k = 6 ∴ P = 6w

Follow-up question If P µ w, and P = 12 when w = 2, find an equation connecting P and w, the value of w when P = 78. (b) When P = 78, 78 = 6w w = 13