 # 5.1.1 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 $\frac{y}{x}=k$  where k is called the constant of variation.

3. y varies directly as x is written as  $y\propto x$ .
4. When $y\propto x$ , 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.

(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:

$\begin{array}{l}\left(a\right)\text{}y=kx,\text{or}\\ \left(b\right)\text{}\frac{{y}_{1}}{{x}_{1}}=\frac{{y}_{2}}{{x}_{2}}\end{array}$

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
$\left(a\right)\text{y}\propto x$
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: $\text{Using}\frac{{y}_{1}}{{x}_{1}}=\frac{{y}_{2}}{{x}_{2}}$
(a) Let x1 = 8 and y1 = 24
$\begin{array}{l}\frac{{y}_{1}}{{x}_{1}}=\frac{{y}_{2}}{{x}_{2}}\to \frac{24}{8}=\frac{{y}_{2}}{{x}_{2}}\\ \frac{3}{1}=\frac{{y}_{2}}{{x}_{2}}\to {y}_{2}=3{x}_{2}\\ \therefore y=3x\end{array}$

(b) Let x1 = 8 and y1 = 24 and x2= 6; find y2.
$\begin{array}{l}\frac{{y}_{1}}{{x}_{1}}=\frac{{y}_{2}}{{x}_{2}}\to \frac{24}{8}=\frac{{y}_{2}}{6}\\ {y}_{2}=\frac{24}{8}\left(6\right)\\ {y}_{2}=18\end{array}$

(c) Let x1 = 8 and y1 = 24 and y2= 36; find x2.
$\begin{array}{l}\frac{{y}_{1}}{{x}_{1}}=\frac{{y}_{2}}{{x}_{2}}\to \frac{24}{8}=\frac{36}{{x}_{2}}\\ 24{x}_{2}=36×8\\ {x}_{2}=12\end{array}$