Question

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

Explanation:

Step1: Identify two points on the line

We can see that the line passes through the point \((0, -4)\) (the y - intercept) and another point, for example, when \(x = 8\), let's find the corresponding \(y\) value. Looking at the line, when \(x = 8\), \(y=-2\)? Wait, no, let's check again. Wait, the line passes through \((0, - 4)\) and let's take another point. Let's use the formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two clear points. The line passes through \((0,-4)\) and \((8, - 2)\)? Wait, no, when \(x = 8\), the \(y\) - coordinate of the line: from the graph, the line at \(x = 8\) is at \(y=-2\)? Wait, no, let's check the grid. Each grid square is 1 unit. Let's take two points: \((0,-4)\) and \((4, - 3)\)? Wait, no, let's do it properly. Let's take \((0,-4)\) and \((8, - 2)\). Wait, the change in \(y\) (\(\Delta y\)) is \(-2-(-4)=2\), and the change in \(x\) (\(\Delta x\)) is \(8 - 0 = 8\). Wait, no, that doesn't seem right. Wait, maybe I made a mistake. Let's take another pair. Let's take \((0,-4)\) and \((4,-3)\). Then \(\Delta y=-3 - (-4)=1\), \(\Delta x = 4-0 = 4\). Wait, no, let's look at the line again. Wait, the line goes from, say, \((-8,-6)\) to \((0,-4)\) to \((8,-2)\). Let's check \((-8,-6)\) and \((0,-4)\). \(\Delta y=-4-(-6)=2\), \(\Delta x=0 - (-8)=8\), so \(m=\frac{2}{8}=\frac{1}{4}\)? Wait, no, wait \((0,-4)\) and \((8,-2)\): \(\Delta y=-2-(-4)=2\), \(\Delta x = 8-0 = 8\), so \(m=\frac{2}{8}=\frac{1}{4}\)? Wait, no, maybe I messed up the points. Wait, let's take \((0,-4)\) and \((4,-3)\). \(\Delta y=-3-(-4)=1\), \(\Delta x = 4 - 0=4\), so \(m=\frac{1}{4}\)? Wait, no, let's check the slope formula again. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: \((0,-4)\) and \((8,-2)\). Then \(y_1=-4\), \(y_2=-2\), \(x_1 = 0\), \(x_2 = 8\). So \(m=\frac{-2-(-4)}{8 - 0}=\frac{2}{8}=\frac{1}{4}\). Wait, but let's check with another pair. Let's take \((-8,-6)\) and \((0,-4)\). \(y_1=-6\), \(y_2=-4\), \(x_1=-8\), \(x_2 = 0\). Then \(m=\frac{-4-(-6)}{0-(-8)}=\frac{2}{8}=\frac{1}{4}\). Yes, that seems consistent.

Step2: Calculate the slope

Using the slope formula \(m = \frac{y_2-y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0,-4)\) and \((x_2,y_2)=(8,-2)\) (or any other two points on the line), we have:
\(m=\frac{-2-(-4)}{8 - 0}=\frac{-2 + 4}{8}=\frac{2}{8}=\frac{1}{4}\)

Answer:

\(\frac{1}{4}\)