To find the solutions to the system of equations
[tex]\[
\begin{cases}
3 x^2 - y^2 = 11 \quad \text{(1)} \\
x^2 + 4 y^2 = 8 \quad \text{(2)}
\end{cases}
\][/tex]
we can use the elimination method step-by-step:
1. Multiply Equation (2) to facilitate elimination:
Multiply Equation (2) by 3 to match the coefficients of [tex]\( x^2 \)[/tex].
[tex]\[
3(x^2 + 4y^2) = 3 \cdot 8
\][/tex]
Simplifying, we get:
[tex]\[
3x^2 + 12y^2 = 24 \quad \text{(3)}
\][/tex]
2. Subtract Equation (1) from Equation (3):
Subtract Equation (1) from Equation (3).
[tex]\[
(3x^2 + 12y^2) - (3x^2 - y^2) = 24 - 11
\][/tex]
Simplifying, we get:
[tex]\[
3x^2 + 12y^2 - 3x^2 + y^2 = 13
\][/tex]
[tex]\[
13y^2 = 13
\][/tex]
[tex]\[
y^2 = 1
\][/tex]
Therefore,
[tex]\[
y = \pm1
\][/tex]
3. Solve for [tex]\( x \)[/tex] when [tex]\( y = 1 \)[/tex] and [tex]\( y = -1 \)[/tex]:
- For [tex]\( y = 1 \)[/tex]:
Substitute [tex]\( y = 1 \)[/tex] into Equation (2):
[tex]\[
x^2 + 4(1)^2 = 8
\][/tex]
Simplifying, we get:
[tex]\[
x^2 + 4 = 8
\][/tex]
[tex]\[
x^2 = 4
\][/tex]
[tex]\[
x = \pm2
\][/tex]
- For [tex]\( y = -1 \)[/tex]:
Substitute [tex]\( y = -1 \)[/tex] into Equation (2):
[tex]\[
x^2 + 4(-1)^2 = 8
\][/tex]
Simplifying, we get:
[tex]\[
x^2 + 4 = 8
\][/tex]
[tex]\[
x^2 = 4
\][/tex]
[tex]\[
x = \pm2
\][/tex]
4. Find all solutions:
The solutions for the system are the pairs [tex]\((x, y)\)[/tex] which combine as:
[tex]\[
(2, 1), (-2, 1), (2, -1), (-2, -1)
\][/tex]
5. Identify positive solutions:
From the solutions, we identify the positive solution for [tex]\( (a, b) \)[/tex]:
[tex]\[
a = 2, \quad b = 1
\][/tex]
6. Answer:
Therefore, the values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
a = 2 \quad \text{and} \quad b = 1
\][/tex]
