Question

solve the following system of equations graphically on the set of axes below. $y = -\frac{1}{2}x + 4$ $y = x - 5$ plot two lines by clicking the graph. click a line to delete it.

Explanation:

Step1: Analyze the first equation \( y = -\frac{1}{2}x + 4 \)

To graph this line, we can find two points. When \( x = 0 \), \( y = -\frac{1}{2}(0) + 4 = 4 \), so one point is \( (0, 4) \). When \( x = 2 \), \( y = -\frac{1}{2}(2) + 4 = -1 + 4 = 3 \), so another point is \( (2, 3) \).

Step2: Analyze the second equation \( y = x - 5 \)

For this line, when \( x = 0 \), \( y = 0 - 5 = -5 \), giving the point \( (0, -5) \). When \( x = 5 \), \( y = 5 - 5 = 0 \), giving the point \( (5, 0) \).

Step3: Find the intersection point

To solve the system graphically, we look for the point where the two lines intersect. By plotting the two lines (using the points found above), we can see that they intersect at \( (6, 1) \). We can also verify this algebraically by setting the two equations equal:
\( -\frac{1}{2}x + 4 = x - 5 \)
\( 4 + 5 = x + \frac{1}{2}x \)
\( 9 = \frac{3}{2}x \)
\( x = 9 \times \frac{2}{3} = 6 \)
Substitute \( x = 6 \) into \( y = x - 5 \), we get \( y = 6 - 5 = 1 \).

Answer:

The solution to the system of equations is \( (6, 1) \)