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 \((2, 2)\) (we can also choose other points, but these are easy to identify).

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 \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting \(x_1 = 0\), \(y_1=-1\), \(x_2 = 2\), and \(y_2 = 2\) into the formula, we get:
\(m=\frac{2-(-1)}{2 - 0}=\frac{2 + 1}{2}=\frac{3}{2}\)? Wait, no, wait. Wait, let's check another point. Wait, maybe I made a mistake. Let's take the point \((1,0)\) and \((0, - 1)\). Then \(x_1=0,y_1 = - 1\), \(x_2=1,y_2=0\). Then \(m=\frac{0-(-1)}{1 - 0}=\frac{1}{1}=1\)? Wait, no, wait the line: when \(x = 0\), \(y=-1\); when \(x = 2\), \(y = 1\)? Wait, maybe I misread the graph. Wait, let's look again. The line crosses the y - axis at \((0,-1)\) and crosses the x - axis at \((1,0)\). So two points are \((0,-1)\) and \((1,0)\).
Now, using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)=(0,-1)\) and \((x_2,y_2)=(1,0)\).
So \(m=\frac{0-(-1)}{1 - 0}=\frac{0 + 1}{1}=\frac{1}{1}=1\)? Wait, no, wait when \(x = 2\), what's \(y\)? Let's see the grid. From \((0,-1)\), if we move 2 units to the right (x increases by 2) and 2 units up (y increases by 2), we get to \((2,1)\). Wait, no, maybe the correct points are \((0,-1)\) and \((2,1)\). Then \(m=\frac{1-(-1)}{2-0}=\frac{2}{2}=1\). Wait, or \((1,0)\) and \((3,2)\). Then \(m=\frac{2 - 0}{3-1}=\frac{2}{2}=1\). So the slope is 1? Wait, no, wait let's check the rise over run. From \((0,-1)\) to \((1,0)\), the rise is \(0-(-1)=1\) and the run is \(1 - 0 = 1\), so slope is \(\frac{1}{1}=1\). Wait, but earlier I thought maybe \(\frac{3}{2}\), but that was a mistake. Let's take another pair: \((-2,-3)\) and \((0,-1)\). Then \(y_2-y_1=-1-(-3)=2\), \(x_2 - x_1=0-(-2)=2\), so \(m=\frac{2}{2}=1\). Yes, so the slope is 1. Wait, maybe I misread the graph initially. Let's confirm: the line goes from \((0,-1)\) up 1 unit and right 1 unit to \((1,0)\), up 1 unit and right 1 unit to \((2,1)\), etc. So the slope is \(\frac{1}{1}=1\). Wait, no, wait the y - intercept is \((0,-1)\) and when \(x = 2\), \(y = 1\), so the change in y is \(1-(-1)=2\), change in x is \(2 - 0=2\), so slope is \(\frac{2}{2}=1\). So the slope is 1.

Wait, maybe my first point selection was wrong. Let's do it properly. Let's take two clear points. The line passes through \((0,-1)\) and \((2,1)\). Then \(y_2 - y_1=1-(-1)=2\), \(x_2 - x_1=2 - 0=2\), so \(m=\frac{2}{2}=1\). Or \((-1,-2)\) and \((1,0)\): \(y_2 - y_1=0-(-2)=2\), \(x_2 - x_1=1-(-1)=2\), so \(m = 1\). So the slope is 1.

Wait, maybe I made a mistake earlier. Let's re - examine the graph. The line: starting from the bottom left, going up, crossing the y - axis at \((0,-1)\), and for every 1 unit we move to the right (in x), we move 1 unit up (in y). So the slope is \(\frac{\text{rise}}{\text{run}}=\frac{1}{1}=1\).

Answer:

\(1\)