To determine the number of solutions to the given system of equations:
[tex]\[
\left\{\begin{array}{l}
x = 5 \\
y = (x - 5)^2 + 6
\end{array}\right.
\][/tex]
we need to follow these steps:
1. Substitute the value of [tex]\( x \)[/tex] from the first equation into the second equation:
- The first equation is [tex]\( x = 5 \)[/tex].
2. Calculate the value of [tex]\( y \)[/tex] using the second equation with the substituted value of [tex]\( x \)[/tex]:
[tex]\[
y = (x - 5)^2 + 6
\][/tex]
Substitute [tex]\( x = 5 \)[/tex] into this equation:
[tex]\[
y = (5 - 5)^2 + 6
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
y = 0^2 + 6
\][/tex]
Further simplify:
[tex]\[
y = 0 + 6
\][/tex]
[tex]\[
y = 6
\][/tex]
3. Identify the solution to the system:
- From the first equation, we know [tex]\( x = 5 \)[/tex].
- From the second equation with the substitution, we have [tex]\( y = 6 \)[/tex].
Therefore, the solution to the system is the ordered pair [tex]\((x, y)\)[/tex]:
[tex]\[
(5, 6)
\][/tex]
4. Determine the number of solutions:
- Since we have found one unique pair [tex]\((x, y)\)[/tex] that satisfies both equations, we can conclude that there is exactly one solution for this system.
Hence, the number of solutions to the given system is:
[tex]\[
\boxed{1}
\][/tex]
