Question

if z varies directly with x and inversely with y , when x=2 , y = 5 and z= 10 , what is z when x =3 and y= 15

Answer 1

Recall:

$\begin{gathered} \text{ Direct Variation is y = }kx \\ \text{ Inverse Variation is y = }\frac{k}{x} \end{gathered}$

If z varies directly with x, that will be z = kx

If z varies inversely with y, that will be z = k/y

Combining the two equations, we get:

$\text{ z = }\frac{\text{ kx}}{\text{ y}}$

We will be using the first set of values to solve for the constant of variation k.

At x = 2, y = 5 and z = 10,

We get,

$\text{ z = }\frac{\text{ kx}}{\text{ y}}$$\text{ 10 = }\frac{2k}{5}$$\text{ }\frac{\text{10 x 5}}{2}\text{ = }k$$\text{ 25 = k}$

Now apply the constant k with the second values to solve for z.

At x = 3 and y = 15,

We get,

$\text{ z = }\frac{\text{ kx}}{\text{ y}}$$\text{ z = }\frac{\text{ 25x}}{\text{ y}}$$\text{ z = }\frac{\text{ 25 x 3}}{\text{ 1}5}$$\text{ z = }\frac{\text{ 7}5}{\text{ 1}5}$$\text{ z = 5}$

Therefore, z = 5