Question

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

Explanation:

Step1: Analyze the green line

The slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. For the green line, the y - intercept ($b$) is - 6 (since it crosses the y - axis at (0, - 6)). To find the slope ($m$), we can use two points. Let's take (0, - 6) and (6, 0). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So $m=\frac{0-(-6)}{6 - 0}=\frac{6}{6}=1$. So the equation of the green line is $y=x - 6$.

Step2: Analyze the purple line

For the purple line, the y - intercept ($b$) is 5? Wait, no, looking at the graph, the purple line crosses the y - axis at (0, 5)? Wait, no, let's re - check. Wait, the purple line: when $x = 0$, $y = 5$? Wait, no, looking at the grid, the purple line passes through (0, 5)? Wait, no, let's take two points. Let's take (0, 5) and (1, 0)? Wait, no, the purple line: from (0, 5) to (1, 0)? Wait, no, the slope calculation: let's take (0, 5) and (1, 0). Wait, no, the purple line: when $x = 0$, $y = 5$? Wait, no, the graph shows the purple line crossing the y - axis at (0, 5)? Wait, no, the original graph: the purple line has a y - intercept at (0, 5)? Wait, no, let's do it properly. Let's take two points on the purple line. Let's say (0, 5) and (1, 0). Wait, the slope $m=\frac{0 - 5}{1-0}=- 5$. Wait, no, maybe (0, 5) and (2, - 5). Then $m=\frac{-5 - 5}{2-0}=\frac{-10}{2}=- 5$. Wait, but let's check the graph again. Wait, the purple line: when $x = 0$, $y = 5$? Wait, no, the user's graph: the purple line crosses the y - axis at (0, 5)? Wait, no, looking at the grid, the y - axis has marks at 8,6,4,2,0,-2,-4,-6,-8. The purple line is at (0, 5)? Wait, no, maybe I made a mistake. Wait, the green line: passes through (0, - 6) and (6, 0), slope is 1, equation $y=x - 6$. The purple line: let's take (0, 5) and (1, 0). No, wait, the purple line: when $x = 0$, $y = 5$? Wait, no, the graph shows the purple line at (0, 5) and then going down. Wait, another way: the purple line, let's take two points. Let's say (0, 5) and (2, - 5). The slope $m=\frac{-5 - 5}{2-0}=\frac{-10}{2}=- 5$. Wait, but maybe the purple line passes through (0, 5) and (1, 0), slope is - 5? Wait, no, let's re - examine. Wait, the purple line: from (0, 5) to (1, 0), the change in y is - 5, change in x is 1, so slope is - 5. So the equation of the purple line is $y=-5x + 5$? Wait, no, that can't be. Wait, maybe I misread the y - intercept. Wait, the purple line: when $x = 0$, $y = 5$? Wait, the grid: the y - axis has 8,6,4,2,0,-2,-4,-6,-8. So the purple line crosses the y - axis at (0, 5)? Wait, no, the original problem's graph: the purple line is at (0, 5)? Wait, no, the user's graph: the purple line is at (0, 5) and then goes down. Wait, but let's check with the green line. The green line is $y=x - 6$. The purple line: let's take (0, 5) and (1, 0), slope is - 5, so equation $y=-5x + 5$? Wait, no, that doesn't seem right. Wait, maybe the purple line has a y - intercept of 5 and slope - 5. Wait, but let's do it again. Let's take two points on the purple line: (0, 5) and (1, 0). Then $m=\frac{0 - 5}{1-0}=- 5$. So the equation is $y=-5x + 5$? Wait, no, when $x = 0$, $y = 5$, correct. When $x = 1$, $y=-5(1)+5 = 0$, correct. So the two equations are:

Green line: $y=x - 6$

Purple line: $y=-5x + 5$? Wait, no, maybe I made a mistake in the y - intercept of the purple line. Wait, looking at the graph again, the purple line crosses the y - axis at (0, 5)? Wait, the grid lines: the y - axis has 8,6,4,2,0,-2,-4,-6,-8. So the point (0, 5) is between 4 and 6. Wait, maybe the purple line's y - intercept is 5? Wait, but let's check the green line again. The green line goes from (0, - 6) to (6, 0), so slope is $\frac{0 - (-6)}{6-0}=1$, so equation $y=x - 6$. The purple line: let's take (0, 5) and (1, 0), slope - 5, equation $y=-5x + 5$. Wait, but maybe the purple line's y - intercept is 5 and slope - 5. Alternatively, maybe the purple line has a y - intercept of 5 and slope - 5, and the green line has a y - intercept of - 6 and slope 1.

Wait, maybe I made a mistake in the purple line's points. Let's take (0, 5) and (2, - 5). Then $m=\frac{-5 - 5}{2-0}=\frac{-10}{2}=- 5$, so equation $y=-5x + 5$. Yes, that works.

Answer:

The system of equations is $y = x-6$ (green line) and $y=-5x + 5$ (purple line)