Question

A daycare facility charges a fee for every minute that a parent is late after the facility’s published closing time. The table represents the fee, y, which is dependent on the number of minutes late, x.

x
y
5
32
8
50
11
68

Which of the following linear equations represents the situation?
y = 3 x + 22
y = 6 x + 36
y = 6 x + 2
y = 24 x + 36

Answer 1

To determine the linear equation representing the situation, we can first analyze the given data points:

1. (5, 32)
2. (8, 50)
3. (11, 68)

Next, we can find the slope (m) of the line given two points. The slope is calculated using the formula:

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

Let's use the points (5, 32) and (8, 50):

\[
m = \frac{50 - 32}{8 - 5} = \frac{18}{3} = 6
\]

Now, we have the slope \( m = 6 \). We can use one of the points to find the y-intercept (b). We'll use the point (5, 32):

Using the slope-intercept form of a linear equation \( y = mx + b \):

\[
32 = 6(5) + b
\]
\[
32 = 30 + b
\]
\[
b = 32 - 30 = 2
\]

Now we can write the equation of the line:

\[
y = 6x + 2
\]

We can check that this equation fits the other points:

For \( x = 8 \):

\[
y = 6(8) + 2 = 48 + 2 = 50 \quad \text{(correct)}
\]

For \( x = 11 \):

\[
y = 6(11) + 2 = 66 + 2 = 68 \quad \text{(correct)}
\]

Thus, the linear equation that represents the situation is:

\[
\boxed{y = 6x + 2}
\]