We are tasked with solving a system of equations:

1. $(x + 5)^2 + (y - 8)^2 = 90$
2. $-2x + y = 3$

Let's solve this step by step.

Step 1: Express $y$ in terms of $x$ using the second equation

From $-2x + y = 3$, we can solve for $y$:
$y = 2x + 3$

Step 2: Substitute $y = 2x + 3$ into the first equation

Substitute $y = 2x + 3$ into $(x + 5)^2 + (y - 8)^2 = 90$:
$(x + 5)^2 + ((2x + 3) - 8)^2 = 90$ Simplify:
$(x + 5)^2 + (2x - 5)^2 = 90$

Expand both squared terms:
$(x^2 + 10x + 25) + (4x^2 - 20x + 25) = 90$

Combine like terms:
$5x^2 - 10x + 50 = 90$

Simplify further:
$5x^2 - 10x - 40 = 0$

Step 3: Solve the quadratic equation

Divide the entire equation by 5 to simplify:
$x^2 - 2x - 8 = 0$

Factorize the quadratic:
$(x - 4)(x + 2) = 0$

Thus, $x = 4$ or $x = -2$.

Step 4: Find the corresponding $y$-values

Using $y = 2x + 3$:

• If $x = 4$:
  $y = 2(4) + 3 = 8 + 3 = 11$ So, one solution is $(4, 11)$.

• If $x = -2$:
  $y = 2(-2) + 3 = -4 + 3 = -1$ So, the other solution is $(-2, -1)$.

Final Answer:

The solutions are:
$\boxed{(4, 11), (-2, -1)}$

Would you like me to verify these solutions or provide additional explanations? Here are 5 related questions to consider:

1. How can you graphically interpret the solution to a nonlinear system?
2. What techniques can be used for solving other types of nonlinear systems?
3. How does substitution simplify solving systems of equations?
4. How can we verify the solutions for this problem?
5. What role do quadratic equations play in this problem-solving approach?

Tip: Always double-check your work by substituting the solutions back into the original equations!