Question

what is the slope of this line?
options: $\frac{3}{4}$, $\frac{4}{3}$, $-\frac{4}{3}$, $-\frac{3}{4}$

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points \((0, -5)\) (the y-intercept) and \((6, 0)\) (where it crosses the x-axis).

Step2: Use the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1) = (0, -5)\) and \((x_2, y_2) = (6, 0)\).
Substitute the values into the formula: \(m = \frac{0 - (-5)}{6 - 0} = \frac{5}{6}\)? Wait, no, maybe I picked the wrong points. Wait, looking again, maybe another pair. Let's take \((0, -5)\) and \((4, -2)\)? Wait, no, let's check the grid. Wait, the line goes through \((0, -5)\) and \((6, 0)\)? Wait, no, when x=6, y=0? Wait, no, the x-intercept is at (6, 0)? Wait, no, looking at the graph, the green line: when x=0, y=-5? Wait, no, maybe I misread. Wait, let's take two clear points. Let's take (0, -5) and (4, -2)? Wait, no, let's use (0, -5) and (6, 0). Wait, the difference in y is 0 - (-5) = 5, difference in x is 6 - 0 = 6? But that's not one of the options. Wait, maybe I made a mistake. Wait, the options are 3/4, 4/3, -4/3, -3/4. Wait, maybe the points are (0, -5) and (4, -2)? No, wait, let's check the grid again. Wait, the line: from (0, -5) to (4, -2): y increases by 3, x increases by 4? No, 0 to 4 is x+4, y from -5 to -2 is y+3. So slope is 3/4? Wait, no, -2 - (-5) = 3, 4 - 0 = 4, so 3/4. Wait, but the options have 3/4. Wait, maybe I messed up the y-intercept. Wait, the line crosses the y-axis at (0, -5)? No, looking at the graph, the y-intercept is at (0, -5)? Wait, no, the grid lines: each square is 1 unit. So from (0, -5) to (4, -2): y goes from -5 to -2 (increase by 3), x from 0 to 4 (increase by 4), so slope is 3/4. Wait, but let's check another pair. From (0, -5) to (6, 0): y increases by 5, x by 6, which is 5/6, not an option. So maybe the correct points are (0, -5) and (4, -2): slope 3/4. Or ( -4, -8) and (0, -5): y increases by 3, x increases by 4, slope 3/4. Yes, that makes sense. So the slope is 3/4? Wait, no, wait, the line is going up from left to right, so positive slope. The options with positive are 3/4 and 4/3. Let's check with (0, -5) and (4, -2): slope is ( -2 - (-5) ) / (4 - 0) = 3/4. Yes, that's one of the options. So the slope is 3/4.
Wait, maybe I made a mistake earlier. Let's confirm. The slope formula is rise over run. From (0, -5) to (4, -2): rise is 3 (from -5 to -2), run is 4 (from 0 to 4), so 3/4. So the slope is 3/4.

Answer:

\(\frac{3}{4}\) (corresponding to the option \(\frac{3}{4}\))