Question

use the elimination method to find all solutions of the system
\\\

$$\begin{cases} 3x^2 - y^2 = 11 \\\\ x^2 + 4y^2 = 8 \\end{cases}$$

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the four solutions of the system are: \\((-a, -b)\\), \\((-a, b)\\), \\((a, -b)\\), and \\((a, b)\\) with positive numbers
\\(a =\\) and \\(b =\\)

Explanation:

Step1: Write down the system of equations

$$
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$$

Step2: Multiply Equation 1 to align coefficients

$$12x^2 - 4y^2 = 44$$

Step3: Add the equations to eliminate \(y^2\)

$$13x^2 = 52$$

Step4: Solve for \(x^2\)

$$x^2 = 4$$

Step5: Substitute \(x^2\) into Equation 2

$$4 + 4y^2 = 8$$

Step6: Solve for \(y^2\)

$$4y^2 = 4 \implies y^2 = 1$$

Step7: Find positive values \(a\) and \(b\)

$$a = \sqrt{4} = 2, \quad b = \sqrt{1} = 1$$

Answer:

\(a = 2\) and \(b = 1\)