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We are given the following system of simultaneous equations:

1. \( x + y = 9 \)
2. \( x^2 - 3xy + 2y^2 = 0 \)

### Step 1: Express \(x\) in terms of \(y\)
From the first equation, solve for \(x\):

\[
x = 9 - y
\]

### Step 2: Substitute \(x = 9 - y\) into the second equation
Now, substitute \(x = 9 - y\) into the second equation:

\[
(9 - y)^2 - 3(9 - y)y + 2y^2 = 0
\]

Expand each term:
\[
(9 - y)^2 = 81 - 18y + y^2
\]
\[
-3(9 - y)y = -27y + 3y^2
\]

Thus, the equation becomes:
\[
(81 - 18y + y^2) + (-27y + 3y^2) + 2y^2 = 0
\]

Simplify:
\[
81 - 18y + y^2 - 27y + 3y^2 + 2y^2 = 0
\]
\[
81 - 45y + 6y^2 = 0
\]

### Step 3: Solve the quadratic equation
We have the quadratic equation:
\[
6y^2 - 45y + 81 = 0
\]

Divide the entire equation by 3 to simplify:
\[
2y^2 - 15y + 27 = 0
\]

Now, solve this quadratic equation using the quadratic formula:
\[
y = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(2)(27)}}{2(2)}
\]
\[
y = \frac{15 \pm \sqrt{225 - 216}}{4}
\]
\[
y = \frac{15 \pm \sqrt{9}}{4}
\]
\[
y = \frac{15 \pm 3}{4}
\]

So, the two solutions for \(y\) are:
\[
y = \frac{15 + 3}{4} = \frac{18}{4} = 4.5
\]
\[
y = \frac{15 - 3}{4} = \frac{12}{4} = 3
\]

### Step 4: Find the corresponding \(x\) values
For \(y = 4.5\), substitute into \(x = 9 - y\):
\[
x = 9 - 4.5 = 4.5
\]

For \(y = 3\), substitute into \(x = 9 - y\):
\[
x = 9 - 3 = 6
\]

### Final Solutions:
The solutions to the system are:
\[
(x, y) = (4.5, 4.5) \quad \text{and} \quad (x, y) = (6, 3)
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to explore further:
1. How can you verify the solutions to a system of simultaneous equations?
2. What are the general methods for solving nonlinear simultaneous equations?
3. Can a system of quadratic equations have more than two solutions?
4. How does the discriminant of a quadratic equation affect the number of solutions?
5. What are the different forms of simultaneous equations, and how do you approach solving each type?

**Tip**: Always check your solutions by substituting them back into both original equations to ensure they satisfy both.

Solve the simultaneous equations: x + y = 9, x^2 - 3xy + 2y^2 = 0

Learn how to solve the simultaneous equations: x + y = 9 and x^2 - 3xy + 2y^2 = 0 with detailed step-by-step solutions. The process involves substitution and the quadratic formula, ideal for high school students in Grades 10-12.

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