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

From the graph, we can see that the line passes through the points \((0, 0)\) and \((1, -8)\) (or we can also use other points like \((0, 0)\) and \((2, -16)\), but let's use \((0, 0)\) and \((1, -8)\) for simplicity). Wait, actually, looking at the y-intercept, when \(x = 0\), \(y = 0\)? Wait, no, maybe I made a mistake. Wait, let's check again. Wait, the line passes through \((0, 0)\) and maybe \((1, -8)\)? Wait, no, let's take two clear points. Let's see, when \(x = 0\), \(y = 0\) (the origin), and when \(x = 1\), what's \(y\)? Wait, looking at the graph, the line goes from, say, \((0, 0)\) to \((1, -8)\)? Wait, no, maybe better to take two points: let's take \((0, 0)\) and \((5, -40)\)? Wait, no, let's check the slope formula. The slope \(m\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: let's say \((0, 0)\) and \((1, -8)\)? Wait, no, maybe \((0, 0)\) and \((2, -16)\)? Wait, no, let's look at the grid. Each grid square is 5 units? Wait, no, the x-axis and y-axis have ticks at -40, -35, -30, ..., 0, 5, 10, ..., 40. Wait, the distance between each grid line is 5 units? Wait, no, the x-axis: from -40 to -35 is 5 units, so each grid square is 5 units? Wait, no, the x-axis labels are -40, -35, -30, ..., 0, 5, 10, ..., 40. So the distance between each major tick (like -40 to -35) is 5 units. So each grid square is 5 units in x and 5 units in y? Wait, no, maybe the grid is 5x5? Wait, no, let's take two points. Let's take (0, 0) and (1, -8)? No, that's not right. Wait, maybe the line passes through (0, 0) and (5, -40)? Wait, no, when x=5, y is -40? Let's check the graph. The line goes from the top left (around x=-5, y=40) to the bottom right (x=5, y=-40). Wait, let's take two points: (0, 0) and (1, -8)? No, that's not. Wait, maybe (0, 0) and (5, -40). Then \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 5\), \(y_2 = -40\). Then slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-40 - 0}{5 - 0}=\frac{-40}{5}=-8\). Wait, that makes sense. Let's verify with another pair. Let's take ( -5, 40) and (0, 0). Then \(x_1=-5\), \(y_1=40\), \(x_2=0\), \(y_2=0\). Then \(m=\frac{0 - 40}{0 - (-5)}=\frac{-40}{5}=-8\). Yes, that works. So the slope is -8.

Wait, maybe I made a mistake earlier. Let's recheck. The slope formula is \(m = \frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: (0, 0) and (1, -8)? No, that's not. Wait, the grid lines: each small square is 5 units? Wait, no, the x-axis has ticks at -40, -35, -30, ..., 0, 5, 10, ..., 40. So the distance between each tick is 5 units. So from x=0 to x=5 is 5 units (1 grid square), and from y=0 to y=-40 is 8 grid squares? No, from y=0 to y=-40 is 8 units of 5? No, wait, y=0 to y=-5 is 1 grid square, y=0 to y=-10 is 2, ..., y=0 to y=-40 is 8 grid squares? No, 40/5=8. So if x increases by 5 (from 0 to 5), y decreases by 40 (from 0 to -40). So \(\Delta x = 5 - 0 = 5\), \(\Delta y = -40 - 0 = -40\). Then slope \(m=\frac{-40}{5}=-8\). Yes, that's correct.

Step2: Calculate the slope using the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points on the line, say \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (5, -40)\). Then:

\(m=\frac{-40 - 0}{5 - 0}=\frac{-40}{5}=-8\)

Answer:

\(-8\)