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 the origin \((0,0)\) and another point, for example, \((4,1)\) (we can also use other points like \((8,2)\), etc.).

Step2: Use the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(4,1)\). Then \(m=\frac{1 - 0}{4 - 0}=\frac{1}{4}\)? Wait, no, let's check another point. Wait, when \(x = 8\), \(y = 2\). So using \((0,0)\) and \((8,2)\), then \(m=\frac{2 - 0}{8 - 0}=\frac{2}{8}=\frac{1}{4}\)? Wait, no, wait the line: when \(x=-4\), \(y = -1\)? Wait no, looking at the graph, the line goes through \((-4,-1)\) and \((4,1)\)? Wait no, let's check the grid. Let's take two clear points. The line passes through \((0,0)\) and \((4,1)\)? Wait no, when \(x = 4\), \(y = 1\)? Wait, no, looking at the graph, the pink line: at \(x = 4\), \(y = 1\); at \(x = 8\), \(y = 2\). So the change in \(y\) is \(2 - 0 = 2\) when change in \(x\) is \(8 - 0 = 8\). So slope \(m=\frac{\Delta y}{\Delta x}=\frac{2}{8}=\frac{1}{4}\)? Wait, no, wait another way: let's take \((-4,-1)\) and \((4,1)\). Then \(\Delta y=1 - (-1)=2\), \(\Delta x=4 - (-4)=8\), so \(m=\frac{2}{8}=\frac{1}{4}\). Wait, but let's check the first point: when \(x = 0\), \(y = 0\); when \(x = 4\), \(y = 1\). So \(\Delta y = 1 - 0 = 1\), \(\Delta x = 4 - 0 = 4\), so \(m=\frac{1}{4}\). Wait, but maybe I made a mistake. Wait, let's take \((-8,-2)\) and \((8,2)\). Then \(\Delta y=2 - (-2)=4\), \(\Delta x=8 - (-8)=16\), so \(m=\frac{4}{16}=\frac{1}{4}\). Yes, that's correct. So the slope is \(\frac{1}{4}\)? Wait, no, wait the line: from \((0,0)\) to \((8,2)\), rise is 2, run is 8, so slope is \(2/8 = 1/4\). Wait, but let's check again. The line: when \(x\) increases by 4, \(y\) increases by 1. So slope is \(1/4\). Wait, but maybe I messed up. Wait, the graph: the pink line, at \(x = -4\), \(y = -1\); at \(x = 0\), \(y = 0\); at \(x = 4\), \(y = 1\); at \(x = 8\), \(y = 2\). So the slope is \(\frac{1 - 0}{4 - 0}=\frac{1}{4}\). Yes, that's correct.

Answer:

\(\frac{1}{4}\)