Let's solve the given system of equations step by step using the elimination method and verify the stated results. The two equations are:

1. $(a+2b)x + (2a-b)y = 2$
2. $(a-2b)x + (2a+b)y = 3$

Step 1: Eliminate $x$

To eliminate $x$, we align the coefficients of $x$ in both equations by multiplying them with appropriate factors. Let us multiply equation (1) by $(a-2b)$ and equation (2) by $(a+2b)$:

$\text{Multiply (1) by } (a-2b): \quad (a-2b)(a+2b)x + (a-2b)(2a-b)y = 2(a-2b)$ $\text{Multiply (2) by } (a+2b): \quad (a+2b)(a-2b)x + (a+2b)(2a+b)y = 3(a+2b)$

Simplify the coefficients of $x$ (since $(a+2b)(a-2b) = a^2 - (2b)^2 = a^2 - 4b^2$):

$(a^2 - 4b^2)x + (a-2b)(2a-b)y = 2(a-2b)$ $(a^2 - 4b^2)x + (a+2b)(2a+b)y = 3(a+2b)$

Now subtract the second equation from the first to eliminate $x$:

[ [(a^2 - 4b^2) - (a^2 - 4b^2)]x + [(a-2b)(2a-b) - (a+2b)(2a+b)]y = 2(a-2b) - 3(a+2b) ]

The $x$-terms cancel out, and we simplify the rest:

$\big[(a-2b)(2a-b) - (a+2b)(2a+b)\big]y = 2(a-2b) - 3(a+2b)$

Step 2: Expand and simplify

Expand $(a-2b)(2a-b)$ and $(a+2b)(2a+b)$:

$(a-2b)(2a-b) = a(2a-b) - 2b(2a-b) = 2a^2 - ab - 4ab + 2b^2 = 2a^2 - 5ab + 2b^2$ $(a+2b)(2a+b) = a(2a+b) + 2b(2a+b) = 2a^2 + ab + 4ab + 2b^2 = 2a^2 + 5ab + 2b^2$

Now subtract: $(2a^2 - 5ab + 2b^2) - (2a^2 + 5ab + 2b^2) = -10ab$

Thus, the left-hand side becomes: $-10ab \cdot y$

Expand and simplify the right-hand side:

$2(a-2b) - 3(a+2b) = 2a - 4b - 3a - 6b = -a - 10b$

So the equation becomes: $-10ab \cdot y = -a - 10b$

Step 3: Solve for $y$

Divide through by $-10ab$: $y = \frac{-a - 10b}{-10ab} = \frac{a + 10b}{10ab}$

Thus, the value of $y$ is: $y = \frac{a + 10b}{10ab}$

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Would you like to proceed with further calculations or explanations? Let me know if anything is unclear! 😊

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Follow-up Questions:

1. Why is elimination a preferred method for solving linear equations in some cases?
2. Can substitution work here as efficiently as elimination? Why or why not?
3. What would happen if $a = 0$ or $b = 0$ in this problem?
4. How does dividing by $-10ab$ affect the sign of the terms in $y$?
5. What is the geometric interpretation of this system of equations?

Tip:

Always check your solution by substituting the values of $x$ and $y$ back into the original equations to verify accuracy.