Let's solve the given system of linear equations step by step:
[tex]\[
\begin{cases}
x + y = 2 \quad \text{(Equation 1)} \\
2x - y = 1 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
### Step 1: Add the equations
First, we add Equation 1 and Equation 2 together to eliminate [tex]\( y \)[/tex]:
[tex]\[
(x + y) + (2x - y) = 2 + 1
\][/tex]
This simplifies to:
[tex]\[
x + y + 2x - y = 3
\][/tex]
[tex]\[
3x = 3
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{3}
\][/tex]
[tex]\[
x = 1
\][/tex]
### Step 2: Substitute [tex]\( x = 1 \)[/tex] into one of the original equations
Now, we substitute [tex]\( x = 1 \)[/tex] into Equation 1 to solve for [tex]\( y \)[/tex]:
[tex]\[
x + y = 2
\][/tex]
[tex]\[
1 + y = 2
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 2 - 1
\][/tex]
[tex]\[
y = 1
\][/tex]
### Solution
Thus, the solution to the system of equations is:
[tex]\[
x = 1, \quad y = 1
\][/tex]
So the final answer is:
[tex]\[
(x, y) = (1, 1)
\][/tex]
