Question

what is the slope of this line? simplify your answer and write it as a proper fraction, improper fraction, or integer.

Explanation:

Step1: Recall the slope formula

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

Step2: Identify the two points

From the graph, we have two points: \((-4, -4)\) and \((4, 2)\). Let \((x_1, y_1)=(-4, -4)\) and \((x_2, y_2)=(4, 2)\).

Step3: Substitute into the slope formula

Substitute the values into the formula: \( m=\frac{2 - (-4)}{4 - (-4)} \).

Step4: Simplify the numerator and denominator

Simplify the numerator: \( 2 - (-4)=2 + 4 = 6 \).
Simplify the denominator: \( 4 - (-4)=4 + 4 = 8 \).

Step5: Reduce the fraction

The fraction \(\frac{6}{8}\) can be reduced by dividing both the numerator and denominator by their greatest common divisor, which is 2. So, \(\frac{6\div2}{8\div2}=\frac{3}{4}\)? Wait, no, wait. Wait, let's check again. Wait, \(2 - (-4)=6\), \(4 - (-4)=8\), so \(\frac{6}{8}=\frac{3}{4}\)? Wait, no, wait, maybe I made a mistake. Wait, let's take another pair of points. Wait, the line also passes through (0, -1) and (4, 2). Let's check with (0, -1) and (4, 2). Then \(y_2 - y_1 = 2 - (-1)=3\), \(x_2 - x_1 = 4 - 0 = 4\). Wait, that's \(\frac{3}{4}\)? Wait, no, wait the first points: (-4, -4) and (4, 2). \(y_2 - y_1 = 2 - (-4)=6\), \(x_2 - x_1 = 4 - (-4)=8\), so \(\frac{6}{8}=\frac{3}{4}\). Wait, but let's check with (0, -1) and (4, 2): \(2 - (-1)=3\), \(4 - 0 = 4\), so \(\frac{3}{4}\). Yes, that's correct. Wait, but wait, maybe I misread the points. Wait, the point (-4, -4) and (4, 2): the difference in y is 2 - (-4)=6, difference in x is 4 - (-4)=8, so 6/8=3/4. Yes.

Wait, but let's check again. Wait, the line goes through (0, -1) and (4, 2). So from (0, -1) to (4, 2), the rise is 3 (from -1 to 2 is 3 units up) and run is 4 (from 0 to 4 is 4 units right), so slope is 3/4. Yes, that's correct. So the slope is 3/4? Wait, no, wait, maybe I made a mistake. Wait, let's take (-4, -4) and (0, -1). Then \(y_2 - y_1 = -1 - (-4)=3\), \(x_2 - x_1 = 0 - (-4)=4\), so 3/4. Yes, so the slope is 3/4.

Wait, but let's do the calculation again with (-4, -4) and (4, 2):

\( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - (-4)}{4 - (-4)}=\frac{2 + 4}{4 + 4}=\frac{6}{8}=\frac{3}{4} \). Yes, that's correct.

Answer:

\(\frac{3}{4}\)