To solve the system of equations, we need to determine whether there is a pair of [tex]\((x, y)\)[/tex] values that satisfies both equations simultaneously. Let's examine the system of equations step by step:
[tex]\[
\left\{\begin{array}{c}
y = x + 11 \\
-y = -x + 11
\end{array}\right.
\][/tex]
First, let's rewrite the second equation to make it easier to compare with the first equation:
[tex]\[
-y = -x + 11
\][/tex]
If we multiply both sides of this equation by [tex]\(-1\)[/tex] to get [tex]\(y\)[/tex] on one side, we obtain:
[tex]\[
y = x - 11
\][/tex]
Now, we have the following two equations:
1. [tex]\( y = x + 11 \)[/tex]
2. [tex]\( y = x - 11 \)[/tex]
Next, we equate the right-hand sides of these equations, since both expressions equal [tex]\( y \)[/tex]:
[tex]\[
x + 11 = x - 11
\][/tex]
Now, let's solve for [tex]\( x \)[/tex]:
[tex]\[
x + 11 = x - 11
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
11 = -11
\][/tex]
This statement, [tex]\( 11 = -11 \)[/tex], is a contradiction. It is impossible for 11 to equal -11. Because we have reached a contradiction, it indicates that there is no pair [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Therefore, the system of equations has no solution.
