Solve the system of linear equations by graphing. \[ \begin{array}{l} y=-\frac{1}{2} x+2 \\ y=\frac{1}{2} x-1 \end{array} \]

Step 1 ：Understand the problem: We are given two linear equations and we are asked to solve the system by graphing. This means we need to graph both equations on the same set of axes and find the point where they intersect. This point of intersection is the solution to the system of equations.

Step 2 ：Graph the first equation: \(y = -\frac{1}{2}x + 2\). This is in slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. The slope is -1/2 and the y-intercept is 2. So we start at the point (0,2) on the y-axis and then move down 1 unit and right 2 units to plot the next point due to the negative slope of -1/2. We continue this pattern to draw the line.

Step 3 ：Graph the second equation: \(y = \frac{1}{2}x - 1\). Again, this is in slope-intercept form. The slope is 1/2 and the y-intercept is -1. So we start at the point (0,-1) on the y-axis and then move up 1 unit and right 2 units to plot the next point due to the positive slope of 1/2. We continue this pattern to draw the line.

Step 4 ：Find the point of intersection: The point where the two lines intersect is the solution to the system of equations. By looking at the graph, we can see that the lines intersect at the point (2,1).

Step 5 ：Check the solution: We can check our solution by substituting the x and y values of the point of intersection into both equations to see if they hold true. For the first equation: \(y = -\frac{1}{2}x + 2\), substitute x = 2 and y = 1, we get \(1 = -\frac{1}{2}*2 + 2\), which simplifies to \(1 = 1\). This holds true. For the second equation: \(y = \frac{1}{2}x - 1\), substitute x = 2 and y = 1, we get \(1 = \frac{1}{2}*2 - 1\), which simplifies to \(1 = 1\). This also holds true.

Step 6 ：Therefore, the solution to the system of equations is \(\boxed{(2,1)}\).