Let’s solve the nonlinear system of equations:

$y = x^2 - 9$ $y = x^2 - 9x$

Step 1: Set the equations equal to each other

Since both equations are equal to $y$, set them equal: $x^2 - 9 = x^2 - 9x$

Step 2: Simplify the equation

Cancel $x^2$ on both sides: $-9 = -9x$

Solve for $x$: $x = 1$

Step 3: Find the corresponding $y$-value

Substitute $x = 1$ into either equation. Using $y = x^2 - 9$: $y = (1)^2 - 9 = -8$

Thus, the solution is: $(1, -8)$

Step 4: Check for additional solutions

Substitute $y = x^2 - 9x$ into $y = x^2 - 9$ again confirms no other $x$-values.

Final Answer:

The solution is $(1, -8)$.

Would you like more explanation or a graph of the solution? Here are some follow-up questions you might consider:

1. How do we handle systems of nonlinear equations geometrically?
2. How can substitution methods help with more complicated systems?
3. Could this system ever have more than one solution? Why or why not?
4. What happens when we use elimination instead of substitution here?
5. How do we solve nonlinear systems involving circles or ellipses?

Tip: Always verify your solution by substituting it back into both equations to ensure correctness.