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 \( y = -\frac{1}{2}x + 2 \)

This is a linear equation in slope - intercept form (\(y=mx + b\)), where the slope \(m =-\frac{1}{2}\) and the y - intercept \(b = 2\). To graph this line, we can find two points. When \(x = 0\), \(y=-\frac{1}{2}(0)+2=2\), so the point \((0,2)\) is on the line. When \(y = 0\), \(0=-\frac{1}{2}x + 2\), then \(\frac{1}{2}x=2\), and \(x = 4\), so the point \((4,0)\) is on the line.

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

We can rewrite it in slope - intercept form. Solve for \(y\): \(y=x - 1\). Here, the slope \(m = 1\) and the y - intercept \(b=-1\). When \(x = 0\), \(y=-1\), so the point \((0, - 1)\) is on the line. When \(y = 0\), \(x-0 = 1\), so \(x = 1\) and the point \((1,0)\) is on the line.

Step3: Find the intersection point

To solve the system graphically, we look for the point where the two lines intersect. We can also solve the system algebraically to verify. From \(x - y=1\), we have \(y=x - 1\). Substitute \(y=x - 1\) into \(y=-\frac{1}{2}x + 2\):
\[

$$\begin{align*} x-1&=-\frac{1}{2}x + 2\\ x+\frac{1}{2}x&=2 + 1\\ \frac{2x+x}{2}&=3\\ \frac{3x}{2}&=3\\ 3x&=6\\ x&=2 \end{align*}$$

\]
Substitute \(x = 2\) into \(y=x - 1\), we get \(y=2 - 1=1\). So the intersection point (the solution of the system) is \((2,1)\).

Answer:

The solution of the system of equations is \((2,1)\)