Question

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

options: a, b, c, d

Explanation:

Step1: Analyze Slopes and Intercepts

For a linear equation \( y = mx + b \), \( m \) is the slope and \( b \) is the y - intercept.

• Look at the two lines in the graph. One line has a positive slope and the other has a negative slope.
• Let's analyze the slopes and y - intercepts of each option:
• Option A: First equation \( y=\frac{2}{5}x - 8 \) (slope \( \frac{2}{5} \), y - intercept - 8), second equation \( y =-\frac{3}{5}x-3 \) (slope \( -\frac{3}{5} \), y - intercept - 3).
• Option B: First equation \( y=\frac{2}{5}x - 3 \) (slope \( \frac{2}{5} \), y - intercept - 3), second equation \( y=-\frac{3}{5}x - 8 \) (slope \( -\frac{3}{5} \), y - intercept - 8).
• Option C: First equation \( y=\frac{5}{2}x - 8 \) (slope \( \frac{5}{2} \), y - intercept - 8), second equation \( y =-\frac{5}{3}x-3 \) (slope \( -\frac{5}{3} \), y - intercept - 3).
• Option D: First equation \( y=\frac{5}{2}x - 3 \) (slope \( \frac{5}{2} \), y - intercept - 3), second equation \( y=-\frac{5}{3}x - 8 \) (slope \( -\frac{5}{3} \), y - intercept - 8).

Step2: Match with Graph

From the graph, one line has a relatively small positive slope (closer to \( \frac{2}{5} \) than \( \frac{5}{2} \)) and a y - intercept of - 3, and the other line has a relatively small negative slope (closer to \( -\frac{3}{5} \) than \( -\frac{5}{3} \)) and a y - intercept of - 8.
So the system of equations \( y=\frac{2}{5}x - 3 \) and \( y =-\frac{3}{5}x-8 \) (Option B) matches the slopes and y - intercepts of the lines in the graph.

Answer:

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