The problem in the image asks you to solve a system of two equations by graphing them. The equations are:

1. $x - 2y = 2$
2. $y = -\frac{1}{2}x + 5$

Let's break this down step by step and show how to graph these equations.

Step 1: Rewrite the first equation in slope-intercept form

The slope-intercept form of an equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Starting with the first equation: $x - 2y = 2$ Solve for $y$: -2y = -x + 2 \quad \text{(subtract x from both sides)} y = \frac{1}{2}x - 1 \quad \text{(divide by -2)}

Now, the first equation is in slope-intercept form: $y = \frac{1}{2}x - 1$

Step 2: Graph both equations

1. First equation: $y = \frac{1}{2}x - 1$

   • Slope $m = \frac{1}{2}$, which means "rise 1, run 2".
   • Y-intercept $b = -1$, meaning the line crosses the y-axis at $(0, -1)$.
   • From the y-intercept, move up 1 unit and to the right 2 units to plot another point.

2. Second equation: $y = -\frac{1}{2}x + 5$

   • Slope $m = -\frac{1}{2}$, which means "fall 1, run 2".
   • Y-intercept $b = 5$, meaning the line crosses the y-axis at $(0, 5)$.
   • From the y-intercept, move down 1 unit and to the right 2 units to plot another point.

Step 3: Find the intersection point

• Graph both lines on the same grid, and the solution to the system will be the point where the two lines intersect. This point of intersection is the solution to the system of equations.

Let me know if you would like more details on how to plot these, or if you have any questions about the graphing process!

Here are five related questions to explore further:

1. How do you determine the slope of a line given its equation in standard form?
2. What are alternative methods to solve a system of equations other than graphing?
3. How can you verify the solution to a system of equations algebraically?
4. What does it mean if two lines are parallel or coincident when graphing systems of equations?
5. How does changing the slope affect the steepness of the graph?

Tip: Always check your points after plotting by substituting them back into the original equations to verify correctness.