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

We can see that the line passes through \((0, -1)\) and \((1, 0)\) (or other clear points like \((5, 3)\) etc.). Let's take \((x_1, y_1) = (0, -1)\) 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 \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Substitute the values: \(y_2 - y_1 = 0 - (-1)=1\) and \(x_2 - x_1 = 1 - 0 = 1\). So \(m=\frac{1}{1}=1\)? Wait, wait, let's check another pair. Let's take \((0, -1)\) and \((5, 3)\). Then \(y_2 - y_1=3 - (-1) = 4\), \(x_2 - x_1=5 - 0 = 5\)? No, that's wrong. Wait, maybe I misread the points. Wait, the line crosses the y - axis at \((0, -1)\) and when \(x = 5\), \(y = 3\)? Wait no, looking at the graph, when \(x = 1\), \(y = 0\); when \(x = 5\), \(y = 3\)? Wait, no, let's recalculate. Wait, the line goes through \((0, -1)\) and \((5, 3)\)? Wait, no, let's take two clear points. Let's take \((0, -1)\) and \((1, 0)\): slope is \(\frac{0 - (-1)}{1 - 0}=\frac{1}{1} = 1\)? Wait, but when \(x = 5\), \(y = 3\)? Wait, \(3-(-1)=4\), \(5 - 0 = 5\)? No, that's not matching. Wait, maybe I made a mistake. Wait, the line passes through \((0, -1)\) and \((1, 0)\), \((2, 1)\), \((3, 2)\), \((4, 3)\), \((5, 4)\)? Wait, maybe my initial point was wrong. Wait, the line: when \(x = 0\), \(y=-1\); when \(x = 1\), \(y = 0\); \(x = 2\), \(y = 1\); \(x = 3\), \(y = 2\); \(x = 4\), \(y = 3\); \(x = 5\), \(y = 4\); \(x = 6\), \(y = 5\); \(x = 7\), \(y = 6\). Ah, I see, I misread the y - value at \(x = 5\). So if we take \((0, -1)\) and \((5, 4)\), then \(y_2 - y_1=4-(-1)=5\), \(x_2 - x_1=5 - 0 = 5\), so \(m=\frac{5}{5}=1\). Wait, but when \(x = 0\), \(y=-1\) and \(x = 1\), \(y = 0\): \(\frac{0 - (-1)}{1 - 0}=\frac{1}{1}=1\). So the slope is 1? Wait, no, wait the line from \((0, -1)\) to \((1, 0)\): rise is 1, run is 1, so slope is 1. Wait, but let's check the y - intercept. Wait, maybe I made a mistake in the y - intercept. Wait, the line crosses the y - axis at \((0, -1)\) and when \(x = 1\), \(y = 0\), so the slope is \(\frac{0 - (-1)}{1 - 0}=1\). Wait, but let's check another pair: \((-1, -2)\) and \((0, -1)\): \(y_2 - y_1=-1-(-2)=1\), \(x_2 - x_1=0 - (-1)=1\), so slope is 1. Yes, so the slope is 1. Wait, but earlier I thought \(x = 5\), \(y = 3\) was wrong, actually at \(x = 5\), \(y = 4\) (since from \(x = 0\), \(y=-1\), each x increase by 1, y increases by 1). So the slope is 1.

Wait, no, wait the graph: let's count the rise over run. From \((0, -1)\) to \((5, 4)\): rise is \(4 - (-1)=5\), run is \(5 - 0 = 5\), so \(5/5 = 1\). From \((0, -1)\) to \((2, 1)\): rise is \(1 - (-1)=2\), run is \(2 - 0 = 2\), so \(2/2 = 1\). So the slope is 1.

Wait, but let's check the line again. The line goes through \((0, -1)\) and \((1, 0)\), so the slope is \(\frac{0 - (-1)}{1 - 0}=1\). So the slope is 1.

Answer:

\(1\)