https://file.aiquickdraw.com/mathbot/file/1726607895828xzc0ossg.jpg

The system of equations in the image is:

\[
x^2 + 9y^2 - 10x + 36y = 20
\]
\[
x - 3y = 2
\]

### Steps to solve the system:

1. **Solve the second equation for one variable** (e.g., solve for \(x\) in terms of \(y\)):

\[
x = 3y + 2
\]

2. **Substitute \(x = 3y + 2\) into the first equation**:

\[
(3y + 2)^2 + 9y^2 - 10(3y + 2) + 36y = 20
\]

Now, let's simplify this expression step by step. I'll expand and simplify the equation first.The solutions for \( y \) are \( y = -2 \) and \( y = 1 \).

Now, let's substitute these values back into the equation \( x = 3y + 2 \) to find the corresponding \( x \) values.

For \( y = -2 \):
\[
x = 3(-2) + 2 = -6 + 2 = -4
\]

For \( y = 1 \):
\[
x = 3(1) + 2 = 3 + 2 = 5
\]

Thus, the solutions to the system are:
1. \( (x, y) = (-4, -2) \)
2. \( (x, y) = (5, 1) \)

Would you like further details on the solution? Here are some related questions you can explore:

1. What happens if the system had more variables?
2. How can this method be generalized for solving nonlinear systems?
3. Can you graph these solutions geometrically?
4. What is the interpretation of these solutions in terms of conic sections?
5. Are there other methods to solve this system of equations?

**Tip:** When solving systems of nonlinear equations, always check if substitution or elimination simplifies the process.

Solve the system of equations: x^2 + 9y^2 - 10x + 36y = 20 and x - 3y = 2.

Learn how to solve a system of nonlinear equations involving both quadratic and linear terms. This problem demonstrates key algebraic methods such as substitution and solving quadratic equations. Ideal for students in Grades 10-12.

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