Question

determine the equation, in slope-intercept form, for the line shown in the graph below:

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -6)\) (the y - intercept) and \((6, 0)\) (the x - intercept). Let \((x_1,y_1)=(0, - 6)\) and \((x_2,y_2)=(6,0)\).

Step2: Calculate the slope \(m\)

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting the values, we get \(m=\frac{0-(-6)}{6 - 0}=\frac{6}{6}=1\)? Wait, no, let's check the points again. Wait, looking at the graph, maybe the y - intercept is \((0,-6)\) and another point: let's take two points. Let's see, when \(x = 0\), \(y=-6\); when \(x = 6\), \(y = 0\)? Wait, no, the slope calculation: \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-6)}{6 - 0}=\frac{6}{6} = 1\)? Wait, but the line seems to have a smaller slope. Wait, maybe I misread the points. Let's take another pair. Let's say when \(x=-6\), what's \(y\)? If the line passes through \((0,-6)\) and \((6,0)\), the slope is 1, but maybe the grid is such that each square is 2 units? Wait, no, the graph has - 6, - 4, - 2, 0, 2, 4, 6 on the x - axis and - 18, - 12, - 6, 0, 6, 12, 18 on the y - axis? Wait, no, the y - axis labels are 18, 12, 6, 0, - 6, - 12, - 18. So each grid square is 3 units? Wait, no, the distance between - 6 and - 4 on the x - axis is 2 units, so each grid square is 1 unit? Wait, maybe I made a mistake. Let's re - examine. Let's take two points: \((0,-6)\) and \((6,0)\). Then the slope \(m=\frac{0 - (-6)}{6-0}=\frac{6}{6}=1\). But the equation of the line in slope - intercept form is \(y=mx + b\), where \(b\) is the y - intercept. The y - intercept \(b=-6\) (since the line crosses the y - axis at \((0,-6)\)). Wait, but let's check with another point. If \(x = 3\), then \(y=m\times3 + b\). If \(m = 1\) and \(b=-6\), then \(y=3-6=-3\). Does that lie on the line? From the graph, it seems so. Wait, but maybe the slope is \(\frac{1}{2}\)? Wait, no, let's recalculate. Wait, maybe the two points are \((0,-6)\) and \((6,0)\). So \(y_2 - y_1=0-(-6)=6\), \(x_2 - x_1=6 - 0 = 6\), so \(m = 1\). Wait, but the line looks like it has a slope of \(\frac{1}{2}\). Wait, maybe I misread the y - intercept. Wait, maybe the y - intercept is \((0,-6)\) and when \(x = 6\), \(y = 0\), so slope is 1. But let's write the equation. Slope - intercept form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We found \(m = 1\) and \(b=-6\), so the equation is \(y=x - 6\)? Wait, no, that can't be. Wait, let's take two points again. Let's say the line passes through \((0,-6)\) and \((6,0)\). Then the slope is \(\frac{0-(-6)}{6 - 0}=1\), so the equation is \(y=x - 6\). But let's check with \(x = 2\), \(y=2-6=-4\). Does that lie on the line? From the graph, it seems plausible. Wait, maybe I was wrong earlier. So the steps:

1. Find the y - intercept (\(b\)): The line crosses the y - axis at \((0,-6)\), so \(b=-6\).
2. Find the slope (\(m\)): Use two points \((x_1,y_1)=(0,-6)\) and \((x_2,y_2)=(6,0)\). \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-6)}{6 - 0}=\frac{6}{6}=1\).
3. Write the equation in slope - intercept form (\(y=mx + b\)): Substitute \(m = 1\) and \(b=-6\) into \(y=mx + b\), we get \(y=x-6\). Wait, but that seems to have a slope of 1. Alternatively, maybe the slope is \(\frac{1}{2}\). Wait, let's check the points again. Let's take \((0,-6)\) and \((4, - 4)\)? No, maybe the grid is such that each square is 2 units. Wait, the x - axis has - 6, - 4, - 2, 0, 2, 4, 6 (difference of 2 between each mark), so each grid square is 2 units? Then the x - coordinate difference between 0 and 6 is 6 units (3 grid squares), and the y - coordinate difference between - 6 and 0 is 6 units (3 grid squares). So slope \(m=\frac{6}{6}=1\) (if each grid square is 2 units, then 3 grid squares is 6 units). So the equation is \(y=x - 6\).

Wait, no, maybe I made a mistake. Let's start over. Slope - intercept form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

1. Find the y - intercept (\(b\)): The line crosses the y - axis at \((0,-6)\), so \(b=-6\).

2. Find two points on the line: Let's take \((0,-6)\) and \((6,0)\) (since when \(x = 6\), \(y = 0\) as per the graph).

3. Calculate the slope \(m\): \(m=\frac{y_2-y_1}{x_2 - x_1}=\frac{0-(-6)}{6 - 0}=\frac{6}{6}=1\).

4. Substitute \(m = 1\) and \(b=-6\) into \(y=mx + b\): \(y=1\times x-6\), so \(y=x - 6\).

Answer:

\(y = x-6\)