To find the equation of the linear function that models the relationship between the number of hours Esperanza works (let's denote this as \(x\)) and the amount of money she earns (let's denote this as \(y\)), we first identify two points based on the information given:
- When \(x = 6\) hours, \(y = 120\) dollars. So the point is \((6, 120)\).
- When \(x = 8\) hours, \(y = 156\) dollars. So the point is \((8, 156)\).
Now, we can calculate the slope \(m\) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points:
\[ m = \frac{156 - 120}{8 - 6} = \frac{36}{2} = 18 \]
Now we know the slope \(m = 18\). Next, we will use one of the points to find the y-intercept \(b\). We can use the point \((6, 120)\):
Using the equation of the line in the form \(y = mx + b\):
\[ 120 = 18(6) + b \]
Calculating \(18 \times 6\):
\[ 120 = 108 + b \]
Now solve for \(b\):
\[ b = 120 - 108 = 12 \]
Now we have both the slope and the y-intercept. The equation of the linear function is:
\[ y = 18x + 12 \]
Thus, the final answer in \(y = mx + b\) form is:
\[ \boxed{y = 18x + 12} \]
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