Question

solve the following system of equations graphically on the set of axes below.
$y = -x + 7$
$y = \frac{1}{4}x - 3$
plot two lines by clicking the graph.
click a line to delete it.

Explanation:

Step1: Find two points for \( y = -x + 7 \)

When \( x = 0 \), \( y = -0 + 7 = 7 \), so we have the point \( (0, 7) \).
When \( x = 7 \), \( y = -7 + 7 = 0 \), so we have the point \( (7, 0) \).

Step2: Find two points for \( y = \frac{1}{4}x - 3 \)

When \( x = 0 \), \( y = \frac{1}{4}(0) - 3 = -3 \), so we have the point \( (0, -3) \).
When \( x = 12 \), \( y = \frac{1}{4}(12) - 3 = 3 - 3 = 0 \), so we have the point \( (12, 0) \).

Step3: Plot the lines and find intersection

Plot the line \( y = -x + 7 \) using \( (0, 7) \) and \( (7, 0) \).
Plot the line \( y = \frac{1}{4}x - 3 \) using \( (0, -3) \) and \( (12, 0) \).
The intersection point of the two lines is the solution. To find it algebraically (to confirm), set \( -x + 7 = \frac{1}{4}x - 3 \).
Multiply both sides by 4: \( -4x + 28 = x - 12 \).
Add \( 4x \) to both sides: \( 28 = 5x - 12 \).
Add 12 to both sides: \( 40 = 5x \), so \( x = 8 \).
Substitute \( x = 8 \) into \( y = -x + 7 \): \( y = -8 + 7 = -1 \).

Answer:

The solution to the system is \( (8, -1) \) (the intersection point of the two lines \( y = -x + 7 \) and \( y = \frac{1}{4}x - 3 \)).