Question

question #5
which graph represents the system shown?
\

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

a
graph a
b
graph b
c
graph c
d
graph d

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 \(b = 0\) (it passes through the origin \((0,0)\) and has a negative slope, so it goes from the second quadrant to the fourth quadrant).

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

In the slope - intercept form \(y=mx + b\), the slope \(m = 2\) (positive slope, so the line is increasing) and the y - intercept \(b = 3\) (the line crosses the y - axis at \((0,3)\)).

Step3: Compare with the graphs

• For graph A: The line \(y = 2x + 3\) has a y - intercept of \(3\) (crosses y - axis at \((0,3)\)) and the line \(y=-x\) passes through the origin with a negative slope. This matches the equations.
• For graph B: The line \(y = 2x+3\) in this graph has a y - intercept of \(2\) (not \(3\)), so it does not match.
• For graph C: The line with negative slope (supposed to be \(y=-x\)) does not pass through the origin, so it does not match.
• For graph D: The line with negative slope (supposed to be \(y=-x\)) does not pass through the origin, and the line \(y = 2x + 3\) has a y - intercept that does not seem to be \(3\) in the context of the other line, so it does not match.

Answer:

A