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 numbers of hours Esperanza works and the amount of money she earns. Enter your answer in y = mx + b form

Answer 1

To find the linear function that models the relationship between the number of hours Esperanza works (x) and the amount of money she earns (y), we need to determine the slope (m) and y-intercept (b) of the line.

We have two points representing Esperanza's work hours and earnings:
1. Point 1: (6 hours, $120) → (6, 120)
2. Point 2: (8 hours, $156) → (8, 156)

### Step 1: Calculate the slope (m)

The formula for the slope \( m \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points we have:
\[
m = \frac{156 - 120}{8 - 6} = \frac{36}{2} = 18
\]

### Step 2: Use the slope-intercept form

The equation of a line is given by:
\[
y = mx + b
\]
Now that we have \( m = 18 \), we can substitute one of the points to find \( b \). Let's use the point (6, 120):

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

### Step 3: Write the final equation

Now, we can combine \( m \) and \( b \) to write the equation of the linear function:

\[
y = 18x + 12
\]

Thus, the equation that models the relationship between the number of hours Esperanza works and the amount of money she earns is:

\[
\boxed{y = 18x + 12}
\]