Question

1. what is the solution to the system of equations graphed below? enter your answer as an ordered pair (x,y)

graph with two lines: y = -3x + 6 and y = 2x - 4 intersecting on a coordinate plane
your answer

Explanation:

Step1: Set the equations equal

Since the solution to a system of linear equations is the point of intersection, we set \( y = -3x + 8 \) equal to \( y = 2x - 4 \):
\( -3x + 8 = 2x - 4 \)

Step2: Solve for x

Add \( 3x \) to both sides:
\( 8 = 5x - 4 \)
Add 4 to both sides:
\( 12 = 5x \)
Divide by 5:
\( x = \frac{12}{5} \)? Wait, no, wait. Wait, maybe I misread the graph. Wait, looking at the graph, the intersection is at x=2? Wait, no, let's solve the equations properly. Wait, the equations are \( y = -3x + 8 \) and \( y = 2x - 4 \). Let's set them equal:

\( -3x + 8 = 2x - 4 \)

Add 3x to both sides: \( 8 = 5x - 4 \)

Add 4 to both sides: \( 12 = 5x \)? No, that can't be. Wait, maybe the equations are \( y = -3x + 6 \) and \( y = 2x - 4 \)? Wait, the first equation in the graph is \( y = -3x + 6 \)? Wait, the user's image shows \( y = -3x + 6 \) (maybe a typo in the original, but let's check the intersection. If we set \( -3x + 6 = 2x - 4 \)):

\( -3x + 6 = 2x - 4 \)

Add 3x to both sides: \( 6 = 5x - 4 \)

Add 4: \( 10 = 5x \)

Divide by 5: \( x = 2 \)

Then plug x=2 into \( y = 2x - 4 \): \( y = 2(2) - 4 = 0 \). So the intersection is at (2, 0). Ah, I see, maybe the first equation was \( y = -3x + 6 \) (since when x=0, y=6, which matches the y-intercept). So let's correct the first equation: \( y = -3x + 6 \) and \( y = 2x - 4 \). Then setting equal:

\( -3x + 6 = 2x - 4 \)

Add 3x: \( 6 = 5x - 4 \)

Add 4: \( 10 = 5x \)

x=2. Then y=2(2)-4=0. So the solution is (2, 0).

Wait, that makes sense. So the ordered pair is (2, 0).

Answer:

(2, 0)