3x^2+3xy+y^2=909 and 3x^2+xy+3y^2=1287 find x+y

We are given the system of equations:

1. \( 3x^2 + 3xy + y^2 = 909 \)
2. \( 3x^2 + xy + 3y^2 = 1287 \)

We need to solve for \( x + y \).

### Step 1: Subtract the equations
We subtract equation (1) from equation (2):

\[
(3x^2 + xy + 3y^2) - (3x^2 + 3xy + y^2) = 1287 - 909
\]

Simplifying the left-hand side:

\[
(3x^2 - 3x^2) + (xy - 3xy) + (3y^2 - y^2) = 1287 - 909
\]

\[
0 - 2xy + 2y^2 = 378
\]

\[
2y^2 - 2xy = 378
\]

Dividing through by 2:

\[
y^2 - xy = 189 \tag{3}
\]

### Step 2: Solve for one variable in terms of the other
We can solve for \( y^2 - xy = 189 \) from equation (3). Let's isolate \( xy \) in this equation:

\[
y^2 - xy = 189 \quad \text{(equation 3)}
\]

We still need to work on combining this with the other equation. Let's try checking for factoring methods or solve one

3x^2 + 3xy + y^2 = 909 and 3x^2 + xy + 3y^2 = 1287, find x + y

Learn how to solve a system of quadratic equations 3x^2 + 3xy + y^2 = 909 and 3x^2 + xy + 3y^2 = 1287 to find x + y. This step-by-step guide covers algebraic manipulation, equation subtraction, and factoring techniques. Suitable for students in Grades 9-12.

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