Question

Yesterday, Esperanza worked for 6 hours and earned $120. Today, Esperanza worked 8 hours and earned $156. Write the equation of the linear function that models the relationship between the number of hours Esperanza works and the amount of money she earns. Enter your answer in y=mx+b
y
=
m
x
+
b
form.

Answer 1

To find the equation of the linear function that models the relationship between the number of hours Esperanza works (x) and the amount of money she earns (y), we can use the two points provided from her work.

From the information given:
- Yesterday (6 hours, $120): This gives us the point (6, 120).
- Today (8 hours, $156): This gives us the point (8, 156).

We can find the slope (m) of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where:
- \((x_1, y_1) = (6, 120)\)
- \((x_2, y_2) = (8, 156)\)

Now plug in the values:

\[
m = \frac{156 - 120}{8 - 6} = \frac{36}{2} = 18
\]

Now that we have the slope \(m = 18\), we can use one of the points to find the y-intercept \(b\). We’ll use the point (6, 120) in the slope-intercept form \(y = mx + b\):

\[
120 = 18(6) + b
\]
\[
120 = 108 + b
\]
\[
b = 120 - 108 = 12
\]

Now we have both \(m\) and \(b\). Therefore, the equation of the linear function is:

\[
y = 18x + 12
\]