Certainly! Let's solve the system of equations to determine the solution:
[tex]\[
\begin{array}{l}
y = -\frac{1}{2} x + 2.5 \\
y = 2 x - 5
\end{array}
\][/tex]
To find the point where these two lines intersect, we need to set the equations equal to each other and solve for [tex]\( x \)[/tex].
1. Set the two equations equal to each other:
[tex]\[
-\frac{1}{2} x + 2.5 = 2 x - 5
\][/tex]
2. Combine like terms to solve for [tex]\( x \)[/tex]:
First, get all [tex]\( x \)[/tex]-terms on one side of the equation and all constants on the other side:
[tex]\[
-\frac{1}{2} x - 2 x = -5 - 2.5
\][/tex]
Simplify the left and right sides:
[tex]\[
-\frac{1}{2} x - 2 x = -\frac{1}{2} x - 2 x = -2.5 x
\][/tex]
[tex]\[
-5 - 2.5 = -7.5
\][/tex]
So we have:
[tex]\[
-2.5 x = -7.5
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-7.5}{-2.5} = 3
\][/tex]
4. Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use the second equation:
[tex]\[
y = 2 x - 5
\][/tex]
[tex]\[
y = 2(3) - 5
\][/tex]
[tex]\[
y = 6 - 5
\][/tex]
[tex]\[
y = 1
\][/tex]
Therefore, the solution to the system of equations is [tex]\((x, y) = (3, 1)\)[/tex].
So, the correct solution is:
[tex]\((3, 1)\)[/tex]
