Question

The table below shows how much Joe earns, y, after working x hours.

Joe’s Earnings

Hours worked
Money earned
4
$30
10
$75
12
$90
22
$165

The relationship between money earned and hours worked is linear. Joe computes the slope between (4, 30) and (12, 90), then computes the slope between (4, 30) and (10, 75). How do the two slopes compare?
The slope between (4, 30) and (12, 90) is greater because the ordered pairs are farther apart on the x-axis.
The slope between (4, 30) and (12, 90) is greater because the ordered pairs are farther apart on the y-axis.
The slope between (4, 30) and (12, 90) and between (4, 30) and (10, 75) is the same.
The slope between (4, 30) and (12, 90) is less because 4 is a factor of 12 and 30 is a factor of 90.

Answer 1

To compare the slopes between the pairs of points, we can use the formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

**First, let's calculate the slope between the points (4, 30) and (12, 90):**

Here, \((x_1, y_1) = (4, 30)\) and \((x_2, y_2) = (12, 90)\).

\[
m = \frac{90 - 30}{12 - 4} = \frac{60}{8} = 7.5
\]

**Next, let's calculate the slope between the points (4, 30) and (10, 75):**

Here, \((x_1, y_1) = (4, 30)\) and \((x_2, y_2) = (10, 75)\).

\[
m = \frac{75 - 30}{10 - 4} = \frac{45}{6} = 7.5
\]

**Conclusion:**

Both calculations yield the same slope of 7.5. This means that the slope between (4, 30) and (12, 90) is the same as the slope between (4, 30) and (10, 75).

Thus, the correct comparison is:

**The slope between (4, 30) and (12, 90) and between (4, 30) and (10, 75) is the same.**