Question

what system of equations does the graph show?
write the equations in slope - intercept form. simplify any fractions.

Explanation:

Step1: Find the equation of the blue line

The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For the blue line, the y-intercept \( b \) is 5 (since it crosses the y-axis at (0, 5)).
To find the slope \( m \), use two points. Let's take (-8, 0) and (0, 5).
The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 0}{0 - (-8)}=\frac{5}{8}\).
So the equation of the blue line is \( y=\frac{5}{8}x + 5 \).

Step2: Find the equation of the purple line

For the purple line, the y-intercept \( b \) is -6 (since it crosses the y-axis at (0, -6)).
To find the slope \( m \), use two points. Let's take (-3, 0) and (0, -6) (wait, actually from the graph, let's check another pair. Let's take (-2, -2) and (0, -6)? Wait, no, let's re-examine. Wait, the purple line goes through (-3, 0)? Wait, no, looking at the graph, the purple line passes through (-3, 0)? Wait, no, when x = -3, y = 0? Wait, no, let's take two points: (-3, 0) and (0, -6). Wait, slope \( m=\frac{-6 - 0}{0 - (-3)}=\frac{-6}{3}=-2 \). Wait, but let's check another point. Wait, the purple line: when x = -1, y = -4? Wait, no, maybe I made a mistake. Wait, the purple line: let's take (0, -6) and (-3, 0). So slope is (0 - (-6))/(-3 - 0)=6/(-3)=-2. So the equation is \( y=-2x - 6 \). Wait, but let's check with x = -2: y = -2*(-2) -6 = 4 -6 = -2. Which matches the graph (the purple line passes through (-2, -2)). Yes, that's correct.

Wait, but let's re-express. Wait, the blue line: when x = -8, y = 0; x = 0, y = 5. So slope is (5 - 0)/(0 - (-8)) = 5/8. So equation \( y=\frac{5}{8}x + 5 \).

The purple line: when x = 0, y = -6; x = -3, y = 0. So slope is (0 - (-6))/(-3 - 0) = 6/(-3) = -2. So equation \( y = -2x - 6 \).

Wait, but let's check the blue line again. Wait, the blue line at x = 0 is y = 5, correct. At x = -8, y = 0, correct. So that's correct.

So the system of equations is:
\( y=\frac{5}{8}x + 5 \)
and
\( y = -2x - 6 \)

Wait, but maybe I made a mistake in the purple line's slope. Let's check another pair. Let's take (-2, -2) and (0, -6). Slope is (-6 - (-2))/(0 - (-2)) = (-4)/2 = -2. Yes, same slope. So the equation is correct.

Answer:

The system of equations is \(

$$\begin{cases} y=\frac{5}{8}x + 5 \\ y=-2x - 6 \end{cases}$$

\)