Question

1. which of the following graphs would fit the system of equations?

y = 2x - 4
y = -\frac{1}{2}x + 1
a)
b)

Explanation:

Step1: Analyze the first equation \( y = 2x - 4 \)

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For \( y = 2x - 4 \), the slope \( m_1 = 2 \) (positive, so the line should rise from left to right) and the y-intercept \( b_1 = -4 \) (the line crosses the y-axis at \( (0, -4) \)).

Step2: Analyze the second equation \( y = -\frac{1}{2}x + 1 \)

For \( y = -\frac{1}{2}x + 1 \), the slope \( m_2 = -\frac{1}{2} \) (negative, so the line should fall from left to right) and the y-intercept \( b_2 = 1 \) (the line crosses the y-axis at \( (0, 1) \)). Also, note that the slopes \( m_1 = 2 \) and \( m_2 = -\frac{1}{2} \) are negative reciprocals (\( 2\times(-\frac{1}{2})=-1 \)), so the lines are perpendicular.

Step3: Evaluate the graphs

• For graph a: We need to check the slopes and y-intercepts. The first line (with positive slope) should have a y-intercept at -4, and the second (negative slope) at 1, and they should be perpendicular.
• For graph b: Check the slopes and y-intercepts. The lines here don't match the y-intercepts (e.g., one line seems to have a y-intercept not at -4 or 1 as needed for our equations).

From the slope and y-intercept analysis, graph a should have one line with positive slope (rising) crossing y at -4 and one with negative slope (falling) crossing y at 1, and they are perpendicular, which matches our equations.

Answer:

a) (the graph labeled a)