Question

1. write the equation of the line in fully simplified slope-intercept form.

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points \((0, -7)\) (the y-intercept) and \((1, -7 + 8)\)? Wait, no, let's check again. Wait, when \(x = 0\), \(y=-7\), and when \(x = 1\), let's see the grid. Wait, actually, let's take two clear points. Let's take \((0, -7)\) and \((1, 1)\)? Wait, no, maybe better to take \((0, -7)\) and \((2, 9)\)? Wait, no, let's calculate the slope. Wait, the slope-intercept form is \(y = mx + b\), where \(b\) is the y-intercept. From the graph, when \(x = 0\), \(y=-7\), so \(b=-7\). Now, let's find the slope \(m\). Let's take two points: \((0, -7)\) and \((1, 1)\)? Wait, no, looking at the graph, when \(x = 1\), what's \(y\)? Wait, the line goes through \((0, -7)\) and \((1, 1)\)? Wait, no, maybe \((0, -7)\) and \((2, 9)\)? Wait, no, let's count the rise over run. From \((0, -7)\) to \((1, 1)\), the rise is \(1 - (-7)=8\), run is \(1 - 0 = 1\), so slope \(m = 8\)? Wait, no, wait, when \(x = 1\), let's check the y-coordinate. Wait, the line at \(x = 1\) is at \(y = 1\)? Wait, no, maybe I made a mistake. Wait, let's look again. The line passes through \((0, -7)\) (since when \(x=0\), \(y=-7\)) and \((1, 1)\)? Wait, no, when \(x = 1\), the y-value is \(1\)? Wait, no, the grid: each square is 1 unit. So from \((0, -7)\) to \((1, 1)\), the change in y is \(1 - (-7)=8\), change in x is \(1 - 0 = 1\), so slope \(m = 8\)? Wait, but let's check another point. When \(x = 2\), \(y = 1 + 8 = 9\), which matches the graph (at \(x = 2\), \(y = 9\)). So the slope \(m = 8\), and the y-intercept \(b=-7\). Wait, no, wait, when \(x = 0\), \(y=-7\), so \(b=-7\). Then the equation is \(y = 8x - 7\)? Wait, no, wait, let's recalculate. Wait, maybe I took the wrong points. Wait, let's take \((0, -7)\) and \((1, 1)\): slope \(m=\frac{1 - (-7)}{1 - 0}=\frac{8}{1}=8\). Then the equation is \(y = 8x - 7\)? Wait, but let's check with \(x = 1\): \(y = 8(1)-7 = 1\), which matches. \(x = 2\): \(y = 16 - 7 = 9\), which also matches. So that's correct.

Wait, but maybe I made a mistake in the y-intercept. Wait, when \(x = 0\), the line crosses the y-axis at \((0, -7)\), so \(b = -7\). Then slope is 8. So the equation is \(y = 8x - 7\).

Step2: Write the equation in slope-intercept form

The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We found \(m = 8\) and \(b = -7\), so the equation is \(y = 8x - 7\).

Answer:

\(y = 8x - 7\)