Question

what system of equations represents the graph? a $y = -\frac{1}{2}x + 10$ $y = -2x$ b $y = -2x + 10$ $y = -\frac{1}{2}x$ c $y = -2x + 10$ $y = -\frac{1}{3}x$ d $y = -\frac{1}{3}x + 10$ $y = -2x$

Explanation:

Step1: Analyze the slopes and intercepts

First, identify the two lines. One line passes through the origin (0,0), so its y-intercept is 0. Let's find its slope. Take two points, say (0,0) and (2, -1) (from the graph). The slope \( m = \frac{-1 - 0}{2 - 0} = -\frac{1}{2} \), so the equation is \( y = -\frac{1}{2}x \).

The other line: let's find its slope and y-intercept. It has a steeper slope. Let's take two points, like (0,10) (y-intercept) and (5,0). The slope \( m = \frac{0 - 10}{5 - 0} = -2 \), so the equation is \( y = -2x + 10 \).

Step2: Match with options

Now check the options. Option B has \( y = -2x + 10 \) (the steeper line) and \( y = -\frac{1}{2}x \) (the less steep line through origin), which matches our analysis.

Answer:

B. \( y = -2x + 10 \)
\( y = -\frac{1}{2}x \)