Question

what is the slope of the line graphed in this grid?
options: $-\frac{3}{2}$, $-\frac{2}{3}$, $\frac{2}{3}$, $\frac{3}{2}$

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -4)\) and \((6, 0)\) (we can also use other points, but these are clear).

Step2: Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Let \((x_1,y_1)=(0, -4)\) and \((x_2,y_2)=(6, 0)\). Then \(m=\frac{0 - (-4)}{6 - 0}=\frac{4}{6}=\frac{2}{3}\). Wait, no, wait. Wait, maybe I picked the wrong points. Wait, looking again, maybe another pair. Let's take \((-6, -8)\) and \((0, -4)\). Then \(y_2 - y_1=-4-(-8) = 4\), \(x_2 - x_1=0 - (-6)=6\), so \(m=\frac{4}{6}=\frac{2}{3}\)? Wait, no, wait the line is going from bottom left to top right, so slope should be positive. Wait, but let's check another pair. Wait, maybe \((3, -2)\) and \((6, 0)\). \(y_2 - y_1=0 - (-2)=2\), \(x_2 - x_1=6 - 3 = 3\), so \(m=\frac{2}{3}\). Wait, but the options have \(\frac{2}{3}\) as an option? Wait, no, wait the options are \(-\frac{2}{3}\), \(-\frac{3}{2}\), \(\frac{2}{3}\), \(\frac{3}{2}\). Wait, maybe I made a mistake. Wait, let's check the y-intercept. The line crosses the y-axis at \((0, -4)\). Let's take another point, say \((3, -2)\). So from \((0, -4)\) to \((3, -2)\), the rise is \(-2 - (-4)=2\), run is \(3 - 0 = 3\), so slope is \(\frac{2}{3}\). So the slope is \(\frac{2}{3}\). Wait, but let me confirm. Wait, the line is increasing, so positive slope. So between \((0, -4)\) and \((6, 0)\), rise is \(0 - (-4)=4\), run is \(6 - 0 = 6\), simplify \(\frac{4}{6}=\frac{2}{3}\). So the slope is \(\frac{2}{3}\).

Answer:

\(\frac{2}{3}\) (the option with \(\frac{2}{3}\))