Let's solve the system of equations given:
- \( x = 1 \)
- \( y = 6 \)
- \( -x + y = 5 \)
- \( y = (x - 3)^2 + 2 \)
From the first two equations, we have \( x = 1 \) and \( y = 6 \).
Let's check if these values satisfy the other two equations.
Checking the third equation: \[ -x + y = 5 \] Substituting \( x = 1 \) and \( y = 6 \): \[ -1 + 6 = 5 \] This equation is satisfied.
Checking the fourth equation: \[ y = (x - 3)^2 + 2 \] Substituting \( x = 1 \): \[ y = (1 - 3)^2 + 2 \] Calculating the right side: \[ y = (-2)^2 + 2 = 4 + 2 = 6 \] This equation is also satisfied.
Since all equations are satisfied by \( x = 1 \) and \( y = 6 \), the solution to the system of equations is: \[ \boxed{(1, 6)} \]