To solve the system of equations using the addition method, we follow these steps:

Given System:

1. $4x^2 - 2y^2 = -4$
2. $3x^2 + 5y^2 = 23$

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Step 1: Eliminate one variable

We aim to eliminate one variable by aligning the coefficients of either $x^2$ or $y^2$. Let’s focus on eliminating $y^2$.

Multiply the equations:

• Multiply the first equation by 5:
  $5(4x^2 - 2y^2) = 5(-4)$
  $20x^2 - 10y^2 = -20$

• Multiply the second equation by 2:
  $2(3x^2 + 5y^2) = 2(23)$
  $6x^2 + 10y^2 = 46$

Now, the system becomes: $20x^2 - 10y^2 = -20 \quad \text{(1)}$ $6x^2 + 10y^2 = 46 \quad \text{(2)}$

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Step 2: Add the equations

Adding equations (1) and (2) cancels out $y^2$: $(20x^2 - 10y^2) + (6x^2 + 10y^2) = -20 + 46$ $26x^2 = 26$ $x^2 = 1$

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Step 3: Solve for $y^2$

Substitute $x^2 = 1$ into one of the original equations. Using $4x^2 - 2y^2 = -4$: $4(1) - 2y^2 = -4$ $4 - 2y^2 = -4$ $-2y^2 = -8$ $y^2 = 4$

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Step 4: Solve for $x$ and $y$

• From $x^2 = 1$, $x = \pm 1$.
• From $y^2 = 4$, $y = \pm 2$.

Thus, the solutions are: $(1, 2), (1, -2), (-1, 2), (-1, -2)$

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Final Solution Set:

The solution set is: $\{ (1, 2), (1, -2), (-1, 2), (-1, -2) \}$

Would you like more details on a specific part? 😊

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Related Questions:

1. What is the substitution method, and how does it compare to the addition method?
2. How do you verify that these solutions satisfy both equations?
3. What are the advantages of the addition method over graphing?
4. How can you solve systems of nonlinear equations graphically?
5. Can the system be solved using matrix methods? If so, how?

Tip: When solving systems of equations, always simplify coefficients to minimize errors during calculations!