Question

write the equation of this line in slope-intercept form.
write your answer using integers, proper fractions, and improper fractions in simplest form.

Explanation:

Step1: Identify slope-intercept form

Slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Find the y - intercept ($b$)

The line crosses the y - axis at $(0, - 7)$, so $b=-7$.

Step3: Calculate the slope ($m$)

We can use two points on the line. Let's take the y - intercept $(0, - 7)$ and another point. From the graph, when $x=-8$, $y = - 1$ (we can see that the line passes through $(-8,-1)$ and $(0, - 7)$). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0, - 7)$ and $(x_2,y_2)=(-8,-1)$. Then $m=\frac{-1-(-7)}{-8 - 0}=\frac{-1 + 7}{-8}=\frac{6}{-8}=-\frac{3}{4}$. Wait, maybe a better pair. Wait, looking at the graph, when $x = 0$, $y=-7$; when $x = 4$, let's see, the line goes down. Wait, maybe I made a mistake. Wait, the line passes through $(0,-7)$ and let's find another point. Let's take $x = - 8$, $y=-1$? Wait, no, when $x=-8$, the y - value: from the graph, the line at $x=-8$ (left side) is at $y=-1$? Wait, no, the line is going from the second quadrant (wait, no, the line is in the third and fourth quadrants). Wait, the line passes through $(0, - 7)$ and let's see, when $x = 4$, what's $y$? Wait, maybe I should take two points: $(0,-7)$ and $(-8, - 1)$? Wait, no, the slope between $(0,-7)$ and $(-8, - 1)$: $\frac{-1-(-7)}{-8-0}=\frac{6}{-8}=-\frac{3}{4}$. Wait, but maybe another pair. Wait, the line goes from $(0,-7)$ and if we move 8 units to the left (x decreases by 8) and y increases by 6? Wait, no, maybe I messed up the points. Wait, let's look again. The line passes through $(0, - 7)$ and when $x=-8$, $y=-1$? Wait, no, the line is going downward from left to right. So as x increases, y decreases. So let's take $(0,-7)$ and $(4, - 10)$? No, maybe better to use the two points: $(0,-7)$ and $(-8, - 1)$ is wrong. Wait, maybe the correct two points are $(0,-7)$ and $(8, - 13)$? No, I think I made a mistake. Wait, let's use the formula correctly. The slope - intercept form is $y=mx + b$, $b = - 7$. Let's pick two points: when $x = 0$, $y=-7$; when $x = 4$, let's see, the line at $x = 4$: from the graph, the line is at $y=-7-\frac{3}{4}(4)=-7 - 3=-10$? No, maybe the correct slope is calculated as follows: the line passes through $(0, - 7)$ and $(-8, - 1)$? Wait, no, the line is going from $(0,-7)$ and if we move 8 units to the left (x=-8), y increases by 6? That would mean slope is $\frac{6}{-8}=-\frac{3}{4}$. But let's check with the graph. Wait, the line is in the third and fourth quadrants. The y - intercept is - 7. Now, let's recalculate the slope. Let's take two points: $(0,-7)$ and $(8, - 13)$? No, that can't be. Wait, maybe I made a mistake in the points. Wait, the line passes through $(0, - 7)$ and let's see, when $x = 8$, what's $y$? The line is going down, so $y$ will be more negative. Wait, maybe the correct two points are $(0,-7)$ and $(-8, - 1)$ is incorrect. Wait, let's use the formula again. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take $(x_1,y_1)=(0,-7)$ and $(x_2,y_2)=(-8, - 1)$. Then $m=\frac{-1-(-7)}{-8 - 0}=\frac{6}{-8}=-\frac{3}{4}$. Wait, but let's check with the equation. If $y=-\frac{3}{4}x-7$, when $x = - 8$, $y=-\frac{3}{4}(-8)-7=6 - 7=-1$, which matches the point $(-8,-1)$. So that works.

Step4: Write the equation

Now that we have $m = -\frac{3}{4}$ and $b=-7$, the equation in slope - intercept form is $y=-\frac{3}{4}x-7$. Wait, but wait, maybe I made a mistake in the points. Wait, let's check another point. If $x = 4$, then $y=-\frac{3}{4}(4)-7=-3 - 7=-10$. Does the line pass through $(4,-10)$? From the graph, it seems so. So the slope is $-\frac{3}{4}$ and y - intercept is - 7.

Wait, but maybe I messed up the slope. Let's try again. The line passes through $(0, - 7)$ and let's take $x = 4$, $y=-10$. Then the slope is $\frac{-10-(-7)}{4 - 0}=\frac{-3}{4}=-\frac{3}{4}$. Yes, that's correct. So the equation is $y = -\frac{3}{4}x-7$.

Answer:

$y = -\frac{3}{4}x - 7$