Answer:
(x, y) ≈ (-2.17, ±sqrt(95)) or (x, y) ≈ (-1.03, ±sqrt(21))
Explanation:
To solve this system of equations, we can use the method of substitution. We can start by isolating one of the variables in one of the equations and substituting it into the other equation. Let's solve for x in the first equation:
5y² + 24x - 77 = 0
24x = 77 - 5y²
x = (77 - 5y²)/24
Now we can substitute this expression for x into the second equation:
14x² + 5y² + 150x + 119 = 0
14((77-5y²)/24)² + 5y² + 150((77-5y²)/24) + 119 = 0
Simplifying this expression gives:
49y^4 - 5390y² - 108090y - 404271 = 0
We can solve for y using the quadratic formula:
y² = (5390 ± sqrt(5390² - 4(49)(-404271)))/(2(49))
y² = (5390 ± sqrt(16946804))/98
y² = (5390 ± 4118)/98
y² = 95 or y² = 21
Substituting each value of y into the expression we found for x earlier gives:
x = (77 - 5(±sqrt(95))²)/24 ≈ -2.17 or x = (77 - 5(±sqrt(21))²)/24 ≈ -1.03
Therefore, the solution to the system of equations is:
(x, y) ≈ (-2.17, ±sqrt(95)) or (x, y) ≈ (-1.03, ±sqrt(21))