To solve the given system of equations:
[tex]\[
\left\{
\begin{array}{c}
y = x + 11 \\
-y = -x + 11
\end{array}
\right.
\][/tex]
we need to determine if there is a common solution for both equations.
First, let's simplify the second equation:
[tex]\[
-y = -x + 11
\][/tex]
Multiply both sides by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = x - 11
\][/tex]
Now we have two equations:
[tex]\[
1. \; y = x + 11
\][/tex]
[tex]\[
2. \; y = x - 11
\][/tex]
To find a solution that satisfies both equations, we can set the right-hand sides of these two expressions for [tex]\( y \)[/tex] equal to each other:
[tex]\[
x + 11 = x - 11
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
11 = -11
\][/tex]
This results in the statement that 11 equals -11, which is clearly false. Since we have reached a contradiction, it indicates that there is no value of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy both equations simultaneously.
Therefore, the solution to this system of equations is:
[tex]\[
\boxed{\text{no solution}}
\][/tex]
