Final answer:
The elimination method involves first aligning the given quadratic equations and then manipulating them to eliminate one variable, followed by solving the resulting quadratic equation to find the values of y and then x while ensuring the solutions are reasonable by checking.
Step-by-step explanation:
To use the elimination method to find all solutions of the system of equations:
- 2x² + 4y = 13
- x² - y² = 7/2
Let's manipulate the first equation to make it more suitable for elimination:
Multiply the second equation by 4:
4x² - 4y² = 14
Now subtract the first equation from this new equation:
(4x² - 4y²) - (2x² + 4y) = 14 - 13
After simplifying, we get:
2x² - 4y² - 2x² - 4y = 1
-4y² - 4y + 1 = 0
Make this a quadratic in terms of y:
4y² + 4y - 1 = 0
Now, solve for y using the quadratic formula where a = 4, b = 4, and c = -1.
After finding the values of y, substitute them back into one of the original equations to find the corresponding x values.
Finally, check the answers to ensure they satisfy both original equations, as this confirms their reasonableness.