Question

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

Explanation:

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

To graph this line, we can find the \( y \)-intercept and the slope. The \( y \)-intercept (\( b \)) is 2, so the point \( (0, 2) \) is on the line. The slope (\( m \)) is \( -\frac{1}{2} \), which means from the \( y \)-intercept, we can go down 1 unit and right 2 units to find another point, e.g., \( (2, 1) \), \( (4, 0) \), etc.

Step2: Analyze the second equation \( x - y = 1 \)

Rewrite it in slope - intercept form (\( y=mx + b \)):
\( x - y=1\) can be rewritten as \( y=x - 1 \).
The \( y \)-intercept is \( - 1 \), so the point \( (0,-1) \) is on the line. The slope is 1, which means from the \( y \)-intercept, we can go up 1 unit and right 1 unit to find another point, e.g., \( (1,0) \), \( (2,1) \), etc.

Step3: Find the intersection point

By plotting the two lines (either by using the points we found or by using a graphing tool), we look for the point where the two lines cross each other.
From the equations, we can also solve the system algebraically to verify. Substitute \( y = x - 1 \) into \( y=-\frac{1}{2}x + 2 \):
\( x - 1=-\frac{1}{2}x+2\)
Add \( \frac{1}{2}x \) to both sides: \( x+\frac{1}{2}x-1 = 2\)
\( \frac{3}{2}x-1=2\)
Add 1 to both sides: \( \frac{3}{2}x=3\)
Multiply both sides by \( \frac{2}{3} \): \( x = 2\)
Substitute \( x = 2 \) into \( y=x - 1 \), we get \( y=2 - 1=1 \)

Answer:

The solution to the system of equations is \( x = 2 \), \( y = 1 \) or the ordered pair \( (2,1) \)