5.1 Direct Variation Part 1

(A) Determining whether a quantity varies directly as another quantity
1. If a quantity varies directly as a quantity x, the
(a) y increases when x increases
(b) y decreases when x decreases
 
2. A quantity 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  y x .
4. When y x , the graph of 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.
y x y = k x 20 = k ( 36 ) k = 20 36 = 4 9 F i n d k f i r s t 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 = k x , or ( b ) y 1 x 1 = y 2 x 2

Example 2

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

Solution:


Method 1: Using y = kx
( a ) y x
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 = 3 x
(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

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