Question

question #5
which graph represents the system shown?
\\(\

$$\begin{cases} x + y = 0 \\\\ y = 2x + 3 \\end{cases}$$

\\)
graphs for options a, b, c, d are shown with coordinate planes and lines. option a: two lines, one with y-intercept 3 and slope 2, another with y-intercept 0 and slope -1. option b: similar lines but with different intercepts? option c: two lines with positive slopes. option d: two lines with positive slopes, one with y-intercept 3 and slope 2, another through origin with positive slope.

Explanation:

Step1: Analyze the first equation \(x + y = 0\)

Rewrite it in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept). We get \(y=-x\). The slope \(m=- 1\) and the y - intercept \(b = 0\), so this line passes through the origin \((0,0)\) and has a negative slope (it goes down from left to right).

Step2: Analyze the second equation \(y = 2x+3\)

In the slope - intercept form \(y=mx + b\), the slope \(m = 2\) (positive slope, goes up from left to right) and the y - intercept \(b = 3\), so this line crosses the y - axis at \((0,3)\).

Step3: Compare with the graphs

• Option A: One line has a y - intercept of \(3\) (matches \(y = 2x + 3\)) and a positive slope, the other line passes through the origin and has a negative slope (matches \(y=-x\)).
• Option B: The line with positive slope has a y - intercept of \(2\) (does not match \(y = 2x+3\) which has a y - intercept of \(3\)).
• Option C: Both lines have positive slopes (the line \(y=-x\) should have a negative slope), so this is incorrect.
• Option D: The line with positive slope has a y - intercept of \(3\), but the other line with positive slope (should be negative for \(y=-x\)) is incorrect.

Answer:

A. The graph in option A, where one line has a y - intercept of 3 and positive slope (for \(y = 2x+3\)) and the other line passes through the origin with a negative slope (for \(y=-x\))