Question

which graph best represents this system of equations and its solution?
2x = 6 - y
5x - 4y = 28
a
graph a
b
graph b
c.
graph c

Explanation:

Step1: Rewrite equations in slope - intercept form ($y = mx + b$)

For the first equation \(2x=6 - y\), we can rewrite it as \(y=-2x + 6\). Here, the slope \(m_1=-2\) and the y - intercept \(b_1 = 6\).
For the second equation \(5x-4y = 28\), we first isolate \(y\):
\(-4y=-5x + 28\)
Divide both sides by \(-4\): \(y=\frac{5}{4}x-7\). Here, the slope \(m_2=\frac{5}{4}\) and the y - intercept \(b_2=-7\).

Step2: Analyze the y - intercepts and slopes

• The first line (\(y = - 2x+6\)) has a y - intercept at \((0,6)\) (positive y - intercept) and a negative slope (the line goes down from left to right).
• The second line (\(y=\frac{5}{4}x - 7\)) has a y - intercept at \((0,-7)\) (negative y - intercept) and a positive slope (the line goes up from left to right).

Now let's analyze the graphs:

• Graph A: Check the y - intercepts and slopes. The first line (with negative slope) should have a y - intercept at \(y = 6\) and the second (positive slope) at \(y=-7\). In graph A, the positive - sloped line seems to have a y - intercept that is not \(-7\).
• Graph B: The line with negative slope has a y - intercept at \(y = 6\) (matches \(y=-2x + 6\)) and the line with positive slope has a y - intercept at \(y=-7\) (matches \(y=\frac{5}{4}x-7\)). Also, the slopes are consistent (negative for the first, positive for the second).
• Graph C: The y - intercepts do not match (the positive - sloped line has a positive y - intercept which should be negative for \(y=\frac{5}{4}x-7\)).

Answer:

B