To solve the given system of equations and find which of the provided solutions satisfies both equations, we must follow these steps:
Given system of equations:
[tex]\[
\begin{cases}
y - 3 = x \\
x^2 - 6x + 13 = y
\end{cases}
\][/tex]
We will check each of the given solution pairs [tex]\((-5, 2)\)[/tex], [tex]\((-2, 1)\)[/tex], [tex]\((2, 5)\)[/tex], and [tex]\((8, 5)\)[/tex] to see which one, if any, satisfies both equations.
1. Checking the pair [tex]\((-5, 2)\)[/tex]:
- Substitute [tex]\(x = -5\)[/tex] and [tex]\(y = 2\)[/tex] into the first equation:
[tex]\[
y - 3 = x \implies 2 - 3 = -5 \implies -1 = -5 \quad (\text{False})
\][/tex]
Since the first equation is not satisfied, [tex]\((-5, 2)\)[/tex] is not a solution.
2. Checking the pair [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = 1\)[/tex] into the first equation:
[tex]\[
y - 3 = x \implies 1 - 3 = -2 \implies -2 = -2 \quad (\text{True})
\][/tex]
- Now check the second equation:
[tex]\[
x^2 - 6x + 13 = y \implies (-2)^2 - 6(-2) + 13 = 1 \implies 4 + 12 + 13 = 1 \implies 29 = 1 \quad (\text{False})
\][/tex]
Since the second equation is not satisfied, [tex]\((-2, 1)\)[/tex] is not a solution.
3. Checking the pair [tex]\((2, 5)\)[/tex]:
- Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 5\)[/tex] into the first equation:
[tex]\[
y - 3 = x \implies 5 - 3 = 2 \implies 2 = 2 \quad (\text{True})
\][/tex]
- Now check the second equation:
[tex]\[
x^2 - 6x + 13 = y \implies 2^2 - 6(2) + 13 = 5 \implies 4 - 12 + 13 = 5 \implies 5 = 5 \quad (\text{True})
\][/tex]
Since both equations are satisfied, [tex]\((2, 5)\)[/tex] is a solution.
4. Checking the pair [tex]\((8, 5)\)[/tex]:
- Substitute [tex]\(x = 8\)[/tex] and [tex]\(y = 5\)[/tex] into the first equation:
[tex]\[
y - 3 = x \implies 5 - 3 = 8 \implies 2 = 8 \quad (\text{False})
\][/tex]
Since the first equation is not satisfied, [tex]\((8, 5)\)[/tex] is not a solution.
After checking all the given pairs, we find that the pair [tex]\((2, 5)\)[/tex] satisfies both equations. Thus, one of the solutions to the given system is [tex]\((2, 5)\)[/tex].
