Question

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 \((0, 8)\) and \((4, 4)\) (we can also use other points, but these are clear). Wait, no, let's check again. Wait, the y-intercept: when \(x = 0\), what's \(y\)? Wait, looking at the graph, when \(x = 4\), \(y = 4\)? Wait, no, maybe I misread. Wait, let's find two clear points. Let's take \((0, 8)\) and \((4, 6)\)? Wait, no, let's do it properly. The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

First, find the y-intercept (\(b\)): when \(x = 0\), the line crosses the y-axis. From the graph, when \(x = 0\), \(y = 8\)? Wait, no, wait the axes: the horizontal axis is y? Wait, no, the labels: the vertical axis is labeled with -12 to 12 (left side) and the horizontal axis (bottom) is x? Wait, no, the graph has the y-axis (horizontal) and x-axis (vertical)? Wait, no, standard coordinate system: horizontal is x, vertical is y. Wait, the labels: the top axis is y (with 12,11,...-12) and the left axis is x (with 12,11,...-12). Wait, that's a bit confusing. Wait, the blue line: let's find two points. Let's take when \(x = 4\) (on the x-axis, vertical), what's y? Wait, no, let's look at the points. Let's see, when \(x = 0\) (vertical axis, x=0), the line is at y=8? Wait, no, maybe the coordinates are (x, y) where x is vertical and y is horizontal. Wait, maybe the graph is rotated? No, standard is x horizontal, y vertical. Wait, the problem says "slope-intercept form" which is \(y = mx + b\), where \(m\) is slope (rise over run, change in y over change in x). Let's find two points. Let's take (x=4, y=8) and (x=0, y=4)? No, that doesn't make sense. Wait, maybe I got the axes wrong. Let's re-express: the horizontal axis (right-left) is y, and vertical axis (up-down) is x. So when x=0 (bottom of vertical axis), y=8 (on horizontal axis). When x=4 (up 4 on vertical axis), y=6 (left 2 on horizontal axis)? Wait, no, let's use two points. Let's take (x=0, y=8) and (x=4, y=6). Wait, the change in y (Δy) is 6 - 8 = -2, change in x (Δx) is 4 - 0 = 4. So slope \(m = \frac{\Delta y}{\Delta x} = \frac{-2}{4} = -\frac{1}{2}\)? No, that doesn't seem right. Wait, maybe the points are (x=0, y=8) and (x= -4, y=10)? No, this is confusing. Wait, let's look at the line: it's going from the bottom left (x negative, y negative?) No, the blue line is going from top right (y positive, x positive) to bottom left (y negative, x negative)? Wait, no, the blue line has arrows: one at bottom left (x negative, y negative?) and one at top right (x positive, y positive). Wait, let's find two clear points. Let's take (x=4, y=4) and (x=0, y=8). Wait, no, the distance between them: from (0,8) to (4,4): Δx = 4 - 0 = 4, Δy = 4 - 8 = -4. So slope \(m = \frac{-4}{4} = -1\)? No, that's not right. Wait, maybe I made a mistake. Wait, let's check the y-intercept. When x=0 (vertical axis, x=0), the line crosses the y-axis (horizontal) at y=8? Wait, no, the y-intercept is where x=0, so (0, b). Let's look at the graph: when x=0 (the vertical line x=0), the line is at y=8? Wait, no, the line passes through (0, 8) and (4, 6)? Wait, no, let's count the grid. Each grid square is 1 unit. Let's take two points: (0, 8) and (4, 6). Wait, the slope would be (6 - 8)/(4 - 0) = -2/4 = -1/2. But then the equation would be \(y = -\frac{1}{2}x + 8\)? But that doesn't seem right. Wait, maybe the points are (0, 8) and (8, 0). Let's check: from (0,8) to (8,0): Δy = 0 - 8 = -8, Δx = 8 - 0 = 8. So slope \(m = \frac{-8}{8} = -1\). Wait, no, 8 - 0 is 8, 0 - 8 is -8, so slope is -1. Then the equation would be \(y = -x + 8\)? Wait, let's test with x=4: y = -4 + 8 = 4. Does the line pass through (4,4)? Let's see, on the graph, when x=4 (vertical axis, x=4), y=4 (horizontal axis, y=4). Yes! So (4,4) is on the line. And (0,8) is on the line (x=0, y=8). So slope \(m = \frac{4 - 8}{4 - 0} = \frac{-4}{4} = -1\). Wait, but then when x=4, y= -4 + 8 = 4, which matches. When x=0, y=8, which matches. Let's check another point: x=8, y= -8 + 8 = 0. Does the line pass through (8,0)? Yes, from the graph, the line crosses the y-axis (horizontal) at y=8 (x=0) and the x-axis (vertical) at x=8 (y=0). So that's correct.

So slope \(m = \frac{0 - 8}{8 - 0} = -1\), y-intercept \(b = 8\).

Step2: Write the slope-intercept form

Slope-intercept form is \(y = mx + b\), where \(m = -1\) and \(b = 8\). So the equation is \(y = -x + 8\). Wait, but let's check another point. When x=4, y= -4 + 8 = 4, which is on the line. When x= -4, y= 4 + 8 = 12, which should be on the line. Yes, that makes sense.

Wait, but earlier I thought the slope was -1, which is correct. So the steps are:

1. Find the y-intercept (b): when x=0, y=8, so b=8.

2. Find the slope (m): using two points (0,8) and (8,0), \(m = \frac{0 - 8}{8 - 0} = \frac{-8}{8} = -1\).

3. Write the equation: \(y = mx + b = -x + 8\).

Answer:

\(y = -x + 8\)