Question

3 a system of equations is graphed on the grid.
which system of equations is best represented by the graph?
a $y = \frac{2}{5}x - 8$
$y = -\frac{3}{5}x - 3$

b $y = \frac{2}{5}x - 3$
$y = -\frac{3}{5}x - 8$

c $y = \frac{5}{2}x - 8$
$y = -\frac{5}{3}x - 3$

d $y = \frac{5}{2}x - 3$
$y = -\frac{5}{3}x - 8$

Explanation:

Step1: Analyze the slope and y-intercept of the first line (positive slope)

The first line (with positive slope) has a y-intercept. From the graph, when \(x = 0\), \(y=-3\) (since it crosses the y-axis at -3). Let's check the slope. For a line \(y = mx + b\), \(m\) is the slope. Let's take two points on the positive - slope line. Let's say when \(x = 5\), \(y=-1\) (from the graph, approximate). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0, - 3)\) and \((x_2,y_2)=(5,-1)\), \(m=\frac{-1-(-3)}{5 - 0}=\frac{2}{5}\). So the equation of the positive - slope line is \(y=\frac{2}{5}x-3\).

Step2: Analyze the slope and y-intercept of the second line (negative slope)

The second line (with negative slope) has a y-intercept. When \(x = 0\), \(y=-8\) (crosses the y-axis at -8). Let's find the slope. Take two points, say \((x_1,y_1)=(0,-8)\) and \((x_2,y_2)=(5,-5)\) (approximate from the graph). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-5-(-8)}{5 - 0}=\frac{3}{5}\), but since the slope is negative, \(m =-\frac{3}{5}\). So the equation of the negative - slope line is \(y=-\frac{3}{5}x - 8\).

Answer:

B. \(y=\frac{2}{5}x - 3\); \(y=-\frac{3}{5}x - 8\)