https://file.aiquickdraw.com/mathbot/file/172636893528053qpzlb7.jpg

The system of equations given in the image is:

\[
\begin{cases}
y = \frac{1}{2}x - 2 \\
-2x + y = 4
\end{cases}
\]

### Step 1: Graph the first equation
The first equation is already in slope-intercept form \( y = mx + b \), where:
- \( m = \frac{1}{2} \) is the slope,
- \( b = -2 \) is the y-intercept.

You can plot the y-intercept at \( (0, -2) \) and use the slope \( \frac{1}{2} \) to rise 1 unit and run 2 units to the right, marking another point at \( (2, -1) \). Draw a line through these points.

### Step 2: Rewrite and graph the second equation
The second equation is \( -2x + y = 4 \). Solve for \( y \) to put it in slope-intercept form:

\[
y = 2x + 4
\]

Here, the slope \( m = 2 \) and the y-intercept \( b = 4 \). Plot the y-intercept at \( (0, 4) \) and use the slope \( 2 \) to rise 2 units and run 1 unit to the right, marking another point at \( (1, 6) \). Draw a line through these points.

### Step 3: Find the solution
To find the solution to this system, we need to find where the two lines intersect. By solving algebraically, set the two expressions for \( y \) equal to each other:

\[
\frac{1}{2}x - 2 = 2x + 4
\]

Multiply both sides by 2 to eliminate the fraction:

\[
x - 4 = 4x + 8
\]

Now solve for \( x \):

\[
x - 4x = 8 + 4
\]
\[
-3x = 12
\]
\[
x = -4
\]

Substitute \( x = -4 \) back into one of the original equations, say \( y = \frac{1}{2}x - 2 \):

\[
y = \frac{1}{2}(-4) - 2 = -2 - 2 = -4
\]

Thus, the solution is \( (-4, -4) \).

This is the point where the two lines intersect, and it is the solution to the system.

### Conclusion:
The solution to the system is \( (-4, -4) \).

Would you like more details on any part of this explanation? Here are 5 related questions you could explore:

1. How do you convert an equation to slope-intercept form?
2. What does the slope of a line represent in real-world contexts?
3. How can you check the solution to a system of equations?
4. What are different methods to solve systems of equations (e.g., substitution, elimination)?
5. How does the graph of a line change when you alter its slope?

**Tip:** When graphing lines, always plot at least two points to ensure accuracy.

Graph the system below and write its solution:

\(
\begin{cases}
y = \frac{1}{2}x - 2 \\
-2x + y = 4
\end{cases}
\)


Learn how to graph and solve the system of linear equations y = (1/2)x - 2 and -2x + y = 4. This problem demonstrates key algebraic concepts including graphing linear equations and finding the point of intersection. Suitable for students in Grades 8-10.

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