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 = }(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 = }(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