To solve the system of linear equations graphically, we need to find where the two lines intersect. Here are the steps:
1. Define the Equations:
We are given two equations:
[tex]\[
y = -\frac{1}{2} x + 3
\][/tex]
[tex]\[
y = \frac{1}{2} x - 2
\][/tex]
2. Set the Equations Equal to Each Other to Find [tex]\( x \)[/tex]:
Since both equations define [tex]\( y \)[/tex], we can set them equal to find the x-coordinate of the intersection:
[tex]\[
-\frac{1}{2} x + 3 = \frac{1}{2} x - 2
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Combine like terms to isolate [tex]\( x \)[/tex]:
[tex]\[
-\frac{1}{2} x + 3 = \frac{1}{2} x - 2
\][/tex]
Add [tex]\(\frac{1}{2} x\)[/tex] to both sides:
[tex]\[
3 = x - 2
\][/tex]
Add 2 to both sides:
[tex]\[
x = 5
\][/tex]
4. Substitute [tex]\( x = 5 \)[/tex] Back into One of the Original Equations to Find [tex]\( y \)[/tex]:
We can use either of the original equations. Using the first one:
[tex]\[
y = -\frac{1}{2} \cdot 5 + 3
\][/tex]
Simplify:
[tex]\[
y = -\frac{5}{2} + 3 = -2.5 + 3 = 0.5
\][/tex]
5. Determine the Point of Intersection:
The point of intersection is [tex]\((5, 0.5)\)[/tex].
6. Evaluate the Possible Answer Choices:
The given answer choices are:
[tex]\[
\left(\frac{1}{2}, -5\right), \left(\frac{1}{2}, 5\right), \left(5, \frac{1}{2}\right), \left(-5, \frac{1}{2}\right)
\][/tex]
Only one of the choices matches our intersection point [tex]\((5, 0.5)\)[/tex].
The correct answer is:
- [tex]\((5, 0.5)\)[/tex], which corresponds to the third choice.
Thus, the correct choice is:
[tex]\[
\boxed{(5, \frac{1}{2})}
\][/tex]
This is the intersection point of the two lines, confirming our calculated solution.
