Let's analyze the given system of equations step-by-step to determine if there is a solution.
The system of equations is:
[tex]\[
\left\{
\begin{array}{l}
y = x + 11 \\
-y = -x + 11
\end{array}
\right.
\][/tex]
First, let's simplify the second equation:
[tex]\[
-y = -x + 11
\][/tex]
Multiply both sides of this equation by [tex]\(-1\)[/tex] to make [tex]\(y\)[/tex] the subject:
[tex]\[
y = x - 11
\][/tex]
Now the system of equations looks like this:
[tex]\[
\left\{
\begin{array}{l}
y = x + 11 \\
y = x - 11
\end{array}
\right.
\][/tex]
Next, we will set the right-hand sides of these two equations equal to each other, because both are equal to [tex]\(y\)[/tex]:
[tex]\[
x + 11 = x - 11
\][/tex]
Subtract [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[
11 = -11
\][/tex]
This results in a contradiction. The statement [tex]\(11 = -11\)[/tex] is never true, which means that there is no value of [tex]\(x\)[/tex] that satisfies both equations simultaneously.
Therefore, the system of equations has no solution.
In conclusion, the answer is:
[tex]\[
\boxed{\text{no solution}}
\][/tex]
