Question

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 \((-6, 0)\) and \((0, 8)\)? Wait, no, let's check the axes. Wait, the x - axis (horizontal) and y - axis (vertical). Wait, looking at the coordinates, let's find two clear points. Let's take the point where the line crosses the x - axis: when \(y = 0\), \(x=-6\)? Wait, no, maybe I mixed up the axes. Wait, the horizontal axis is y - axis? Wait, no, the standard is x - horizontal, y - vertical. Wait, in the graph, the horizontal axis has labels from - 8 to 8 (left - right) and vertical axis from - 8 to 8 (up - down). Wait, the line passes through \((-8, - 4)\) and \((0, 8)\)? No, let's look at the grid. Wait, another approach: the slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's find two points. Let's take \((-6,0)\) and \((0,8)\)? No, wait, when \(x=-6\), \(y = 0\) (on the x - axis), and when \(x = 0\), \(y=8\) (on the y - axis)? Wait, no, the line goes from the bottom left to the top right. Let's take two points: let's say \((-8, - 4)\) and \((0, 8)\). Wait, no, let's check the grid. Wait, the line passes through \((-6,0)\) (x - intercept) and \((0,8)\) (y - intercept)? Wait, no, the slope calculation: let's take two points. Let's take \((-6,0)\) and \((0,8)\). Then \(y_2 - y_1=8 - 0 = 8\), \(x_2 - x_1=0-(-6)=6\). So slope \(m=\frac{8}{6}=\frac{4}{3}\)? Wait, no, maybe I got the axes wrong. Wait, the horizontal axis is y - axis? Wait, the labels: the horizontal axis (right - left) has labels from - 8 to 8 (y - axis), and vertical axis (up - down) has labels from - 8 to 8 (x - axis). Oh! That's the mistake. So the horizontal axis is y - axis (so the variable is y), and vertical axis is x - axis (variable is x). So the coordinates are (x, y), where x is vertical, y is horizontal. So a point is (x, y) with x on the vertical axis, y on the horizontal axis. So let's find two points. Let's take the point where x = - 4 (vertical) and y=-8 (horizontal), and another point where x = 8 (vertical) and y = 0 (horizontal). Wait, no, the line passes through (x1,y1)=(-8, - 4) and (x2,y2)=(0, 8)? No, let's look at the line: when y = - 6 (horizontal), x = - 4 (vertical); when y = 0 (horizontal), x = 0 (vertical); when y = 6 (horizontal), x = 4 (vertical). Wait, so two points: (x1,y1)=(-4, - 6) and (x2,y2)=(4, 6). Then \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{6-(-6)}{4-(-4)}=\frac{12}{8}=\frac{3}{2}\)? No, wait, no, the slope is \(\frac{\Delta x}{\Delta y}\) if we consider x as vertical and y as horizontal? Wait, no, the standard slope is \(\frac{\Delta y}{\Delta x}\), but here the axes are swapped? Wait, the problem is to find the slope of the line. Let's correct the axes: in the graph, the horizontal axis (left - right) is the y - axis (so y - values), and the vertical axis (up - down) is the x - axis (x - values). So a point is (x, y) where x is the vertical coordinate, y is the horizontal coordinate. So let's find two points on the line. Let's take (x1,y1)=(-4, - 6) and (x2,y2)=(4, 6). Then the change in y is \(y_2 - y_1=6-(-6)=12\), change in x is \(x_2 - x_1=4-(-4)=8\). Wait, no, slope is \(\frac{\text{change in }x}{\text{change in }y}\)? No, no, slope is \(\frac{\text{change in }y}{\text{change in }x}\) when x is horizontal and y is vertical. But here, the horizontal axis is y (so y is horizontal), vertical axis is x (x is vertical). So the slope formula is \(m=\frac{\Delta x}{\Delta y}\)? Wait, no, let's think of the line as a function of y: x = f(y). Then the slope (rate of change of x with respect to y) is \(\frac{\Delta x}{\Delta y}\). Let's take two points: when y=-6, x=-4; when y = 6, x = 4. Then \(\Delta x=4-(-4)=8\), \(\Delta y=6-(-6)=12\). So \(m=\frac{8}{12}=\frac{2}{3}\)? No, that doesn't seem right. Wait, maybe the axes are standard: horizontal is x - axis, vertical is y - axis. Let's re - check the graph. The horizontal axis (left - right) has labels from - 8 to 8 (x - axis), vertical axis (up - down) from - 8 to 8 (y - axis). Then the line passes through (-6,0) (x - intercept) and (0,8) (y - intercept). Then slope \(m=\frac{8 - 0}{0-(-6)}=\frac{8}{6}=\frac{4}{3}\). Wait, but let's take another pair of points. Let's take (-3,4) and (3,8). Then \(y_2 - y_1=8 - 4 = 4\), \(x_2 - x_1=3-(-3)=6\), so \(m=\frac{4}{6}=\frac{2}{3}\)? No, I'm confused. Wait, the correct way: find two points with integer coordinates. Let's look at the line: it passes through (-6,0) and (0,8). So using slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{8 - 0}{0-(-6)}=\frac{8}{6}=\frac{4}{3}\). Wait, but maybe the points are (-8, - 4) and (0, 8). Then \(y_2 - y_1=8-(-4)=12\), \(x_2 - x_1=0-(-8)=8\), so \(m=\frac{12}{8}=\frac{3}{2}\). Wait, now I'm really confused. Wait, the graph: the line goes from the bottom left (where x is negative, y is negative) to the top right (x positive, y positive). Let's count the rise over run. From (-6,0) to (0,8): rise is 8 (from y = 0 to y = 8), run is 6 (from x=-6 to x = 0). So rise over run is 8/6 = 4/3. Yes, that makes sense. So the slope is 4/3.

Step2: Calculate the slope

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)=(-6,0)\) and \((x_2,y_2)=(0,8)\). Then \(y_2 - y_1=8 - 0 = 8\), \(x_2 - x_1=0-(-6)=6\). So \(m=\frac{8}{6}=\frac{4}{3}\).

Answer:

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