Question

e the following system of equations graphically on the set of axes below.
$y = -x + 4$
$y = 2x - 2$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find points for \( y = -x + 4 \)

Substitute \( x = 0 \): \( y = 0 + 4 = 4 \) (point \( (0, 4) \)).
Substitute \( x = 4 \): \( y = -4 + 4 = 0 \) (point \( (4, 0) \)).

Step2: Find points for \( y = 2x - 2 \)

Substitute \( x = 0 \): \( y = 0 - 2 = -2 \) (point \( (0, -2) \)).
Substitute \( x = 1 \): \( y = 2 - 2 = 0 \) (point \( (1, 0) \)).

Step3: Solve for intersection (optional, to verify)

Set \( -x + 4 = 2x - 2 \).
Simplify: \( 3x = 6 \Rightarrow x = 2 \).
Substitute \( x = 2 \) into \( y = -x + 4 \): \( y = 2 \).
Intersection: \( (2, 2) \).

Plot the points for each line on the graph to visualize the system.

Answer:

To graph the system, we find points for each line:

For \( y = -x + 4 \):

• When \( x = 0 \), \( y = 4 \) (point: \( (0, 4) \))
• When \( x = 4 \), \( y = 0 \) (point: \( (4, 0) \))

For \( y = 2x - 2 \):

• When \( x = 0 \), \( y = -2 \) (point: \( (0, -2) \))
• When \( x = 1 \), \( y = 0 \) (point: \( (1, 0) \))

Plot these points and draw the lines. The intersection point (solution) is found by solving the system:
Set \( -x + 4 = 2x - 2 \)
\( 4 + 2 = 2x + x \)
\( 6 = 3x \)
\( x = 2 \)
Substitute \( x = 2 \) into \( y = -x + 4 \): \( y = -2 + 4 = 2 \)
So the solution is \( (2, 2) \).