Q38. - Algebra 1

Question

Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve.

Suppose you earn $127 for working 20 hours.

a. Write a direct variation equation relating your earning to the number of hours worked.

b. How much would you earn for working 35 hours?

Step-by-Step Solution

Answer

a. The direct variation equation relating the earning to the number of hours worked is $y=\frac{127}{20}x$, where y represents the earning and x represents the number of hours worked.

b. The amount earned after working for 35 hours is 222.25.

Part a. Step 1. Write the general equation of direct variation.

Let y represents the earning and x represents the number of hours worked.

It is given that earning varies directly to the number of hours worked.

Therefore, it implies that $y\alpha x$.

Therefore, it is obtained that:

$\begin{array}{l}y\alpha x\\ y=kx\end{array}$

Where k is constant of proportionality.

Part a. Step 2. Write a direct variation equation that relates the earning to the number of hours worked .

It is given that the earning earned after working for 20 hours is $127.

That implies, it is given that when $x=20, y=127$.

Therefore, substitute 20 for x and 127 for y in the equation $y=kx$ to find the value of k.

$\begin{array}{c}y=kx\\ 127=k\left(20\right)\\ \frac{127}{20}=k\end{array}$

Substitute the value of k in the equation $y=kx$.

Therefore, it is obtained that:

$\begin{array}{c}y=kx\\ y=\frac{127}{20}x\end{array}$

Therefore, the direct variation equation relating the earning to the number of hours worked is $y=\frac{127}{20}x$, where y represents the earning and x represents the number of hours worked.

Part b. Step 1. Find the value of y when x = 35 .

The direct variation equation that relates x and y is $y=\frac{127}{20}x$.

Find the value of y by substituting 35 for x in the equation $y=\frac{127}{20}x$.

$\begin{array}{c}y=\frac{127}{20}x\\ =\frac{127}{20}\left(35\right)\\ =222.25\end{array}$

Therefore, the value of y when $x=35$ is 222.25.

Part b. Step 2. Find the amount earned after working for 35 hours.

As, y represents the earning and x represents the number of hours worked.

 

Therefore, in order to find the amount earned after working for 35 hours, the value of y is to be calculated when $x=35$.

 

The value of y when $x=35$ is 222.25.

 

Therefore, the amount earned after working for 35 hours is 222.25.