# Assume y varies directly as x. If x= 2 when y= 7, find x when y= 32.

## Question:

Assume y varies directly as x. If x= 2 when y= 7, find x when y= 32.

## Direct Variation:

Two variables change directly if an increase in one variable increases the other variable and if its decrease reduces the other variable. For variables that are directly related, their ratio is equal to a constant which is known as the constant of variation.

If the variable {eq}y {/eq} varies directly as {eq}x {/eq}, we can write this relationship as:

• {eq}y\propto x {/eq}.

Replacing the proportionality constant with an equal sign and adding a variation constant:

• {eq}y = k x {/eq}.

If {eq}x = 2 {/eq} when {eq}y = 7 {/eq}, then the proportionality constant is equal to:

• {eq}7 = k\times 2 {/eq}.
• {eq}k = \dfrac{7}{2} = 3.5 {/eq}.

Thus, the equation relating {eq}x\; and\; y {/eq} is:

• {eq}y = 3.5x {/eq}.

Using the equation above, the value of {eq}x {/eq} when {eq}y = 32 {/eq} is equal to:

• {eq}32 = 3.5x {/eq}.
• {eq}x = \dfrac{32}{3.5} = \boxed{\dfrac{64}{7}} {/eq}. 