5.1 Direct Variation Part 1

(A) Determining whether a quantity varies directly as another quantity

1. If a quantity y varies directly as a quantity x, the
(a) y increases when x increases
(b) y decreases when x decreases

2. A quantity y varies directly as a quantity x if and only if   y x =k where k is called the constant of variation.

3. y varies directly as x is written as  yx .

4. When yx , the graph of y against x is a straight line passing through the origin.


(B) Expressing a direct variation in the form of an equation involving two variables

Example 1
Given that y varies directly as x and y = 20 when x = 36 . Write the direct variation in the form of equation.

yx y=kx 20=k(36) k= 20 36 = 4 9 Find k first y= 4 9 x

(C) Finding the value of a variable in a direct variation

1. When y varies directly as x and sufficient information is given, the value of y or x can be determined by using:

(a) y=kx, or (b)  y 1 x 1 = y 2 x 2


Example 2
Given that y varies directly as x and y = 24 when x = 8, find
(a) The equation relating y to x
(b) The value of y when = 6
(c) The value of x when = 36

Solution:

Method 1: Using y = kx
(a) yx
y = kx
when y = 24, x = 8
24 = k (8)
k = 3
y = 3x

(b) when x = 6,
y = 3(6)
y = 18

(c) when y = 36
36 = 3x
x =12


Method 2: Using  y 1 x 1 = y 2 x 2   
(a) Let x1 = 8 and y1 = 24
y 1 x 1 = y 2 x 2 24 8 = y 2 x 2 3 1 = y 2 x 2 y 2 =3 x 2 y=3x
(b) Let x1 = 8 and y1 = 24 and x2 = 6; find y2.
y 1 x 1 = y 2 x 2 24 8 = y 2 6 y 2 = 24 8 (6) y 2 =18
(c) Let x1 = 8 and y1 = 24 and y2 = 36; find x2.
y 1 x 1 = y 2 x 2 24 8 = 36 x 2 24 x 2 =36×8 x 2 =12