Question

the image shows a coordinate plane with two lines: $y = \frac{1}{3}x - 8$ and $y = -\frac{3}{2}x + 3$. the lines are graphed on a grid with x and y axes labeled with numbers from -10 to 10. the equations of the lines are also written to the right of the graph: $y = -\frac{3}{2}x + 3$ and $y = \frac{1}{3}x - 8$.

Explanation:

Step1: Analyze the equation \( y = -\frac{3}{2}x + 3 \)

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For \( y = -\frac{3}{2}x + 3 \), the slope \( m = -\frac{3}{2} \) (negative, so the line should decrease from left to right) and the y-intercept \( b = 3 \) (the line crosses the y-axis at \( (0, 3) \)).

Step2: Analyze the equation \( y = \frac{1}{3}x - 8 \)

For \( y = \frac{1}{3}x - 8 \), the slope \( m = \frac{1}{3} \) (positive, so the line should increase from left to right) and the y-intercept \( b = -8 \) (the line crosses the y-axis at \( (0, -8) \)).

Step3: Match with the graph

Looking at the graph, the line labeled \( y = -\frac{3}{2}x + 3 \) has a negative slope (decreasing) and crosses the y-axis at \( (0, 3) \), which matches the equation's properties. The other line \( y = \frac{1}{3}x - 8 \) has a positive slope (increasing) and a y-intercept at a negative value, which also matches its equation. But if we are identifying the line for \( y = -\frac{3}{2}x + 3 \), we check the slope and intercept. The line with the equation \( y = -\frac{3}{2}x + 3 \) in the graph is the one with the negative slope and y-intercept at 3.

Answer:

The line representing \( y = -\frac{3}{2}x + 3 \) is the one with the equation labeled \( y = -\frac{3}{2}x + 3 \) (or visually, the line with negative slope crossing the y-axis at 3). If we consider the two lines, the one with equation \( y = -\frac{3}{2}x + 3 \) is the correct match for its equation.