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 \((0, -1)\) and \((1, 0)\) (we can also choose other points, but these are easy to identify). Let \((x_1, y_1) = (0, -1)\) and \((x_2, y_2) = (1, 0)\).

Step2: Use the slope formula

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Substitute the values of the points into the formula:
\(m=\frac{0 - (-1)}{1 - 0}=\frac{0 + 1}{1}=\frac{1}{1} = 1\)? Wait, no, let's check another pair of points. Let's take \((1,0)\) and \((2,1)\). Then \(m=\frac{1 - 0}{2 - 1}=\frac{1}{1}=1\)? Wait, or \((0, -1)\) and \((2,1)\): \(m=\frac{1 - (-1)}{2 - 0}=\frac{2}{2}=1\)? Wait, no, wait the y-intercept is at \((0, -1)\) and when x=1, y=0; x=2, y=1; x=3, y=2, etc. Wait, actually, let's take two points: (0, -1) and (3, 2). Then \(y_2 - y_1=2 - (-1)=3\), \(x_2 - x_1=3 - 0 = 3\), so \(m=\frac{3}{3}=1\)? Wait, no, wait the line goes from (0, -1) up to (1, 0), so the rise is 1, run is 1, so slope is 1? Wait, but let's check again. Wait, when x=0, y=-1; x=1, y=0; x=2, y=1; x=3, y=2. So the change in y (Δy) between (0, -1) and (3, 2) is 2 - (-1)=3, change in x (Δx) is 3 - 0=3, so slope is 3/3=1. Wait, but let's check the point (0, -1) and ( - 3, -4). Then Δy=-4 - (-1)=-3, Δx=-3 - 0=-3, so slope is (-3)/(-3)=1. So the slope is 1? Wait, no, wait maybe I made a mistake. Wait, the line passes through (0, -1) and (1, 0), so slope is (0 - (-1))/(1 - 0)=1/1=1. Wait, but let's check another way. The equation of the line: y = mx + b, where b is the y-intercept. From the graph, when x=0, y=-1, so b=-1. So the equation is y = mx -1. When x=1, y=0, so 0 = m(1) -1 → m=1. So the slope is 1. Wait, but let's check with (2,1): y=1*2 -1=1, which matches. (3,2): 3 -1=2, matches. So the slope is 1. Wait, but wait the line in the graph: from (0, -1) to (1, 0) is up 1, right 1, so slope is 1. So the slope is 1. Wait, but maybe I misread the graph. Wait, the y-axis: at x=0, y is -1, and at x=1, y is 0, x=2, y=1, etc. So the slope is (0 - (-1))/(1 - 0)=1/1=1. So the slope is 1.

Wait, but let's check the two points (0, -1) and (1, 0). So Δy = 0 - (-1) = 1, Δx = 1 - 0 = 1, so slope is 1/1 = 1. So the slope is 1.

Answer:

1