Question

what is the slope of the line that passes through the points in this table?

x	y
3	2
7	5
11	8
15	11

options: $-\frac{4}{3}$, $-\frac{3}{4}$, $\frac{3}{4}$, $\frac{4}{3}$

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Choose two points

Let's take the first two points \((3, 2)\) and \((7, 5)\). Here, \( x_1 = 3 \), \( y_1 = 2 \), \( x_2 = 7 \), \( y_2 = 5 \).

Step3: Calculate slope

Substitute into the formula: \( m = \frac{5 - 2}{7 - 3} = \frac{3}{4} \)? Wait, no, wait. Wait, let's check another pair. Take \((7, 5)\) and \((11, 8)\). \( x_1 = 7 \), \( y_1 = 5 \), \( x_2 = 11 \), \( y_2 = 8 \). Then \( m = \frac{8 - 5}{11 - 7} = \frac{3}{4} \)? Wait, no, wait, 8 - 5 is 3, 11 - 7 is 4. Wait, but let's check the first and third points: \((3, 2)\) and \((11, 8)\). \( m = \frac{8 - 2}{11 - 3} = \frac{6}{8} = \frac{3}{4} \). Wait, but wait, the x-values are 3,7,11,15 (difference of 4) and y-values are 2,5,8,11 (difference of 3). So slope is \( \frac{\Delta y}{\Delta x} = \frac{3}{4} \)? Wait, no, wait, 5 - 2 is 3, 7 - 3 is 4. So \( \frac{3}{4} \). Wait, but let's confirm. The slope formula is rise over run, which is \( \frac{y_2 - y_1}{x_2 - x_1} \). So with points (3,2) and (7,5): \( y_2 - y_1 = 5 - 2 = 3 \), \( x_2 - x_1 = 7 - 3 = 4 \), so slope is \( \frac{3}{4} \).

Answer:

\(\frac{3}{4}\) (corresponding to the option with \(\frac{3}{4}\))