__5.3 Joint Variation__

**5.3a Representing a Joint Variation using the symbol**

**‘α’.**

**1.**If one quantity is proportional to two or more other quantities, this relationship is known as

**joint variation**.

**2.**‘

*y*varies directly as

*x*and

*z*’ is written as

*y*α

*xz*.

**3.**‘

*y*varies directly as

*x*and inversely

*z*’ is written as $y\text{}\alpha \text{}\frac{x}{z}.$

**4.**‘

*y*varies inversely as

*x*and

*z*’ is written as $y\text{}\alpha \text{}\frac{1}{xz}.$

Example 1:

Example 1:

State the relationship of each of the following variations using the symbol ‘α’.

**(a)**

*x*varies jointly as

*y*and

*z*.

**(b)**

*x*varies inversely as

*y*and $\sqrt{z}.$

**(c)**

*x*varies directly as

*r*

^{3}and inversely as

*y*.

*Solution:***5.3b Solving Problems involving Joint Variation**

**1.**If $y\text{}\alpha \text{}{x}^{n}{z}^{n},\text{then}y=k{x}^{n}{z}^{n}$ , where

*k*is a constant and

*n*= 2, 3 and ½.

**2.**If $y\text{}\alpha \text{}\frac{1}{{x}^{n}{z}^{n}},\text{then}y=\frac{1}{k{x}^{n}{z}^{n}}$ , where

*k*is a constant and

*n*= 2, 3 and ½.

**3.**If $y\text{}\alpha \text{}\frac{{x}^{n}}{{z}^{n}},\text{then}y=\frac{k{x}^{n}}{{z}^{n}}$ , where

*k*is a constant and

*n*= 2, 3 and ½.

**Example 2:**

Given that
$p\text{}\alpha \text{}\frac{1}{{q}^{2}\sqrt{r}}$
when

*p*= 4,*q*= 2 and*r*= 16, calculate the value of*r*when*p*= 9 and*q*= 4.

*Solution:*