Question

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

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identify two points

We have two points: \((0, -1)\) and \((-6, -6)\). Let \((x_1, y_1)=(0, -1)\) and \((x_2, y_2)=(-6, -6)\).

Step3: Substitute into slope formula

Substitute the values into the formula: \( m=\frac{-6 - (-1)}{-6 - 0} \). Simplify the numerator: \(-6 - (-1)=-6 + 1=-5\). Simplify the denominator: \(-6 - 0=-6\). Wait, maybe we should take another pair of points for accuracy. Let's take \((0, -1)\) and \((6, 4)\) (from the graph, when \(x = 6\), \(y = 4\)). Now, \((x_1, y_1)=(0, -1)\), \((x_2, y_2)=(6, 4)\). Then \( m=\frac{4 - (-1)}{6 - 0}=\frac{4 + 1}{6}=\frac{5}{6}\)? Wait, no, wait the point \((-6, -6)\): let's recalculate with \((-6, -6)\) and \((0, -1)\). \(y_2 - y_1=-1 - (-6)=-1 + 6 = 5\). \(x_2 - x_1=0 - (-6)=0 + 6 = 6\). Ah, I had the order wrong. So slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1 - (-6)}{0 - (-6)}=\frac{5}{6}\)? Wait, no, wait the correct way: slope is rise over run, change in y over change in x. Let's take two clear points: \((0, -1)\) and \((6, 4)\). Change in y: \(4 - (-1)=5\). Change in x: \(6 - 0 = 6\). Wait, but also, \((-6, -6)\) and \((0, -1)\): change in y is \(-1 - (-6)=5\), change in x is \(0 - (-6)=6\). So slope is \(\frac{5}{6}\)? Wait, no, wait the point \((0, -1)\) and when \(x = 6\), \(y = 4\): yes, \(4 - (-1)=5\), \(6 - 0 = 6\), so slope is \(\frac{5}{6}\)? Wait, no, maybe I made a mistake. Wait, let's check with \((-6, -6)\) and \((0, -1)\): \(y\) increases by \(5\) (from \(-6\) to \(-1\)) when \(x\) increases by \(6\) (from \(-6\) to \(0\)). So slope is \(\frac{5}{6}\)? Wait, no, wait the graph: when \(x = 0\), \(y=-1\); when \(x = 6\), \(y = 4\). So the change in y is \(4 - (-1)=5\), change in x is \(6 - 0 = 6\), so slope is \(\frac{5}{6}\)? Wait, but let's check another pair: \((-6, -6)\) and \((6, 4)\). Change in y: \(4 - (-6)=10\), change in x: \(6 - (-6)=12\), simplify \(\frac{10}{12}=\frac{5}{6}\). Yes, so the slope is \(\frac{5}{6}\)? Wait, no, wait I think I messed up the points. Wait the point \((0, -1)\) and \((6, 4)\): yes, that's correct. So slope is \(\frac{4 - (-1)}{6 - 0}=\frac{5}{6}\). Wait, but let's check the first point \((-6, -6)\): from \((-6, -6)\) to \((0, -1)\), y goes from -6 to -1 (increase by 5), x goes from -6 to 0 (increase by 6), so slope is 5/6.

Wait, maybe I made a mistake earlier in the order. So the correct slope is \(\frac{5}{6}\)? Wait, no, wait the graph: the line passes through (0, -1) and when x=6, y=4? Wait, no, looking at the graph, when x=6, y=3? Wait, no, the grid: each square is 1 unit. Let's re-examine the graph. The point (0, -1) is on the y-axis. Then, moving to the right 6 units (x=6), what's the y-coordinate? Let's count the grid. From (0, -1), moving up 5 units and right 6 units: -1 + 5 = 4, x=0 + 6=6. So (6, 4) is on the line. Then another point: (-6, -6): from (0, -1), moving left 6 units (x=-6) and down 5 units (y=-1 -5=-6). So yes, the two points are (-6, -6) and (6, 4), or (0, -1) and (6, 4), or (-6, -6) and (0, -1). So the slope is (change in y)/(change in x) = (4 - (-6))/(6 - (-6))=10/12=5/6, or (-1 - (-6))/(0 - (-6))=5/6, or (4 - (-1))/(6 - 0)=5/6. So the slope is \(\frac{5}{6}\)? Wait, no, wait I think I see the mistake. Wait the point (0, -1) and (6, 4): 4 - (-1)=5, 6 - 0=6, so 5/6. But let's check with (0, -1) and (1, y): when x=1, y should be -1 + m. If m=5/6, then y=-1 + 5/6=-1/6, but the graph seems to pass through (1, 0)? Wait, no, the graph passes through (1, 0)? Wait, (0, -1), (1, 0): then slope is (0 - (-1))/(1 - 0)=1/1=1? Wait, now I'm confused. Wait the graph: the line passes through (0, -1) and (1, 0)? Let's check the grid. The y-axis is at x=0, y=-1. Then at x=1, y=0? Because from (0, -1), moving right 1, up 1: (1, 0). Then at x=2, y=1; x=3, y=2; x=4, y=3; x=5, y=4; x=6, y=5? Wait, that's different. Oh! I misread the graph. Let's look again. The red line: at x=0, y=-1. At x=1, y=0 (since it crosses the x-axis at (1, 0)? Wait, no, the point (0, -1) is on the y-axis. Then, when x=1, y=0? Let's count the grid. Each square is 1 unit. So from (0, -1) to (1, 0): change in y is 1, change in x is 1, so slope is 1? Wait, but the other point (-6, -6): from (-6, -6) to (0, -1): change in y is 5, change in x is 6, so 5/6. But if it passes through (1, 0), then slope is 1. There's a contradiction here. Wait, let's look at the graph again. The line: starting from the bottom left, going through (-6, -6), (0, -1), and then up. Wait, when x=0, y=-1; x=6, y=4? No, maybe the grid is such that each square is 1 unit, and the line passes through (0, -1) and (6, 4) is wrong. Wait, (0, -1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5). Ah! There we go. So (0, -1), (1, 0): slope is (0 - (-1))/(1 - 0)=1. Then (6, 5): (5 - (-1))/(6 - 0)=6/6=1. Oh! I see, I misread the y-coordinate at x=6. It's 5, not 4. So the point (6, 5), not (6, 4). So that's the mistake. So let's recalculate with (0, -1) and (6, 5). Change in y: 5 - (-1)=6. Change in x: 6 - 0=6. So slope is 6/6=1. Also, with (-6, -6) and (0, -1): change in y is -1 - (-6)=5? No, wait (0, -1) and (6, 5): change in y is 5 - (-1)=6, change in x is 6 - 0=6, so slope 1. And (-6, -6) and (0, -1): change in y is -1 - (-6)=5? No, wait (-6, -6) to (0, -1): y goes from -6 to -1, which is +5, x goes from -6 to 0, which is +6. But (0, -1) to (6, 5): y goes from -1 to 5, +6, x goes from 0 to 6, +6. So slope is 6/6=1. Ah, I see, I misread the y-coordinate at x=6. It's 5, not 4. So the correct slope is 1. Let's check with (1, 0): (0, -1) to (1, 0): change in y=1, change in x=1, slope=1. Yes, that makes sense. So the error was in misreading the y-value at x=6. So the correct two points are (0, -1) and (6, 5), or (0, -1) and (1, 0), or (-6, -6) and (0, -1) (but wait, (-6, -6) to (0, -1): y change is 5, x change is 6, but that's wrong because (0, -1) to (6, 5) is y change 6, x change 6. So maybe the point (-6, -6) is not on the line? Wait, the graph shows a point at (-6, -6) and (0, -1). Let's check the slope between (-6, -6) and (0, -1): (-1) - (-6)=5, 0 - (-6)=6, so 5/6. But between (0, -1) and (6, 5): 5 - (-1)=6, 6 - 0=6, so 1. There's a contradiction, which means I misread the points. Let's look at the graph again. The line passes through (0, -1) (marked) and (-6, -6) (marked). Let's count the grid: from (0, -1) to (-6, -6): left 6 units (x from 0 to -6), down 5 units (y from -1 to -6). So change in y is -5, change in x is -6. So slope is (-5)/(-6)=5/6. But also, from (0, -1) to (6, 4): right 6 units, up 5 units, slope 5/6. And from (6, 4) to (12, 9): slope 5/6. So maybe the point (6, 5) is not on the line, but (6, 4) is. So my initial reading was correct, and the (1, 0) is not on the line. Let's confirm with the two marked points: (0, -1) and (-6, -6). So using the slope formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1)=(0, -1)\) and \((x_2, y_2)=(-6, -6)\). Then \(m=\frac{-6 - (-1)}{-6 - 0}=\frac{-5}{-6}=\frac{5}{6}\). Alternatively, \((x_1, y_1)=(-6, -6)\) and \((x_2, y_2)=(0, -1)\), then \(m=\frac{-1 - (-6)}{0 - (-6)}=\frac{5}{6}\). So the slope is \(\frac{5}{6}\)? Wait, but when I look at the graph, from (0, -1), moving right 6 units (x=6), y should be -1 + (5/6)*6 = -1 + 5 = 4, which matches (6, 4). So (6, 4) is on the line. And (1, -1 + 5/6)= (1, -1/6), which is not (1, 0), so (1, 0) is not on the line. So the correct slope is 5/6? Wait, now I'm really confused. Let's use the two marked points: (0, -1) and (-6, -6). So:

\(x_1 = 0\), \(y_1 = -1\)

\(x_2 = -6\), \(y_2 = -6\)

Slope \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - (-1)}{-6 - 0} = \frac{-5}{-6} = \frac{5}{6}\)

Yes, that's correct. So the slope is \(\frac{5}{6}\).

Answer:

\(\frac{5}{6}\)