To find the solution for the system of linear equations:

1. $y = -\frac{1}{2}x + 2$
2. $y = \frac{1}{4}x + 2$

we need to find the point where these two lines intersect.

Step 1: Set the equations equal to each other

Since both equations are equal to $y$, we can set them equal to each other:

$-\frac{1}{2}x + 2 = \frac{1}{4}x + 2$

Step 2: Eliminate the constants

Subtract 2 from both sides:

$-\frac{1}{2}x = \frac{1}{4}x$

Step 3: Solve for $x$

To eliminate the fractions, multiply every term by 4:

$4 \times -\frac{1}{2}x = 4 \times \frac{1}{4}x$

This simplifies to:

$-2x = x$

Now, add $2x$ to both sides:

$-2x + 2x = x + 2x$

Simplifying this gives:

$0 = 3x$

Dividing by 3:

$x = 0$

Step 4: Solve for $y$

Substitute $x = 0$ into one of the original equations (e.g., $y = -\frac{1}{2}x + 2$):

$y = -\frac{1}{2}(0) + 2 = 2$

Final Answer

The solution to the system of equations is the point where they intersect:

$(x, y) = (0, 2)$

Would you like more details or have any questions? Here are five related questions:

1. How can we verify that $(0, 2)$ is indeed the point of intersection?
2. What do the slopes of these lines tell us about their relationship?
3. How would the solution change if the constant terms were different?
4. What does the intersection point represent graphically?
5. How can we solve this system using a graphing method?

Tip: When solving systems of equations, setting the equations equal to each other is a common method for finding the point of intersection.