To solve the given system of equations and determine which pair(s) satisfy both equations, let's go through the steps systematically for each given pair.
The equations are:
1. [tex]\( y - 3 = x \)[/tex]
2. [tex]\( x^2 - 6x + 13 = y \)[/tex]
Given pairs are [tex]\((-5, 2)\)[/tex], [tex]\((-2, 1)\)[/tex], [tex]\((2, 5)\)[/tex], and [tex]\((8, 5)\)[/tex].
### Checking each pair:
1. For [tex]\((-5, 2)\)[/tex]:
- Substitute [tex]\( x = -5 \)[/tex] and [tex]\( y = 2 \)[/tex] into the first equation:
[tex]\[
y - 3 = 2 - 3 = -1 \quad \text{which is not equal to} \quad -5
\][/tex]
This pair does not satisfy the first equation.
2. For [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 1 \)[/tex] into the first equation:
[tex]\[
y - 3 = 1 - 3 = -2 \quad \text{which is equal to} \quad -2
\][/tex]
This pair satisfies the first equation.
- Substitute [tex]\( x = -2 \)[/tex] into the second equation:
[tex]\[
x^2 - 6x + 13 = (-2)^2 - 6(-2) + 13 = 4 + 12 + 13 = 29
\][/tex]
[tex]\[
y = 1 \quad \text{which is not equal to} \quad 29
\][/tex]
This pair does not satisfy the second equation.
3. For [tex]\((2, 5)\)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 5 \)[/tex] into the first equation:
[tex]\[
y - 3 = 5 - 3 = 2 \quad \text{which is equal to} \quad 2
\][/tex]
This pair satisfies the first equation.
- Substitute [tex]\( x = 2 \)[/tex] into the second equation:
[tex]\[
x^2 - 6x + 13 = 2^2 - 6(2) + 13 = 4 - 12 + 13 = 5
\][/tex]
[tex]\[
y = 5 \quad \text{which is equal to} \quad 5
\][/tex]
This pair satisfies the second equation.
4. For [tex]\((8, 5)\)[/tex]:
- Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 5 \)[/tex] into the first equation:
[tex]\[
y - 3 = 5 - 3 = 2 \quad \text{which is not equal to} \quad 8
\][/tex]
This pair does not satisfy the first equation.
### Conclusion:
After checking all the given pairs, we found that the pair [tex]\((2, 5)\)[/tex] satisfies both equations:
1. [tex]\( y - 3 = 2 \)[/tex]
2. [tex]\( x^2 - 6x + 13 = 5 \)[/tex]
Thus, one of the solutions to the system of equations is: [tex]\((2, 5)\)[/tex].
