Question

Yesterday, Esperanza worked fir 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 form.(1 point)

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 can use the two sets of data provided.

1. From yesterday's work:
- Hours (x₁) = 6
- Earnings (y₁) = 120

2. From today's work:
- Hours (x₂) = 8
- Earnings (y₂) = 156

Next, we can calculate the slope (m) of the linear function using the formula:

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

Substituting the values:

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

Now we know the slope \( m = 18 \).

Next, we can use one of the points to find the y-intercept (b). We'll use the point from yesterday (6 hours, $120).

The equation in the point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( (x_1, y_1) = (6, 120) \) and \( m = 18 \):

\[
y - 120 = 18(x - 6)
\]

Distributing the 18:

\[
y - 120 = 18x - 108
\]

Now, adding 120 to both sides:

\[
y = 18x + 12
\]

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

\[
y = 18x + 12
\]

Where:
- \( m = 18 \) (the rate of earnings per hour)
- \( b = 12 \) (the y-intercept, indicating a fixed amount or base pay)