Question

question 14. which graph represents the system of equations below? \\(\

$$\begin{cases} -2x + y = 9 \\\\ \\frac{1}{2}x + y = 4 \\end{cases}$$

\\) a. graph

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation \(-2x + y=9\), we can rewrite it as \(y = 2x+9\). The slope of this line is \(m_1 = 2\) (a positive slope, so the line should be increasing) and the y - intercept is \(b_1=9\) (the line crosses the y - axis at \((0,9)\)).

For the second equation \(\frac{1}{2}x + y=4\), we rewrite it as \(y=-\frac{1}{2}x + 4\). The slope of this line is \(m_2=-\frac{1}{2}\) (a negative slope, so the line should be decreasing) and the y - intercept is \(b_2 = 4\) (the line crosses the y - axis at \((0,4)\)).

Step2: Analyze the graph

We need to find a graph where one line has a positive slope (increasing) with a y - intercept of 9 and another line has a negative slope (decreasing) with a y - intercept of 4.

Looking at the given graph option A, we check the slopes and y - intercepts. The steeper line with a positive slope should correspond to \(y = 2x + 9\) (since slope 2 is steeper than slope \(-\frac{1}{2}\)) and the less - steep, decreasing line should correspond to \(y=-\frac{1}{2}x + 4\). The y - intercept of the increasing line should be 9 (high on the y - axis) and the y - intercept of the decreasing line should be 4.

Answer:

A. (The graph labeled A, which has one increasing line with a relatively high y - intercept and one decreasing line with a y - intercept of 4, matching the slopes and y - intercepts of the two equations \(y = 2x+9\) and \(y=-\frac{1}{2}x + 4\))