# Solve: If y is proportional to x and x is 4 when y is 22, then what is y when: (g)x =...

## Question:

Solve: If y is proportional to x and x is 4 when y is 22, then what is y when: {eq}(g)x = \frac{1}{w} {/eq}?

y =

## Equations and Proportionality:

Suppose A is in direct proportion of B then we'll write the expression as {eq}A\propto B {/eq}. To eliminate the direct proportionality constant, we need to add a proportionality constant and make an equation such as {eq}A=k B {/eq} and then simplify the equation according to the given conditions.

The given data are:

• The relation between the variables {eq}x {/eq} and {eq}y {/eq} is {eq}y\propto x {/eq}.
• The values of the variables for the first condition is {eq}x=4\\ y=22 {/eq}.
• The value of the variable {eq}x {/eq} for the second condition is {eq}x = \displaystyle \frac{1}{w} {/eq}.

Simplifying the given direct proportion of the variables, we get:

{eq}y=kx...................(1) {/eq}

Plug {eq}x=4 {/eq} and {eq}y=22 {/eq} in the above equation and simplify it for the proportionality constant {eq}k {/eq}.

{eq}\begin{align*} y&=kx\\ 22&=k(4)\\ k&=\displaystyle \frac{22}{4} \end{align*} {/eq}

According to the second condition, plug {eq}k=\displaystyle \frac{22}{4} {/eq} and {eq}x = \displaystyle \frac{1}{w} {/eq} in equation (1) and simplify it.

{eq}\begin{align*} y&=\displaystyle \left ( \frac{22}{4}\right )\left ( \frac{1}{w} \right )\\ &=\boxed{\displaystyle \frac{11}{2w}} \end{align*} {/eq}