Answer: bellow!
Step-by-step explanation:
(5)/(x^(2)) - (2)/(y^(2)) + 3 = 0 ---(1)
(3)/(x^(2)) + (1)/(y^(2)) = 7 ---(2)
Let's solve it using the method of substitution.
From equation (1), we can rearrange it to isolate y:
(5)/(x^(2)) + 3 = (2)/(y^(2))
(2)/(y^(2)) = (5)/(x^(2)) + 3
To make it easier, let's rewrite (5)/(x^(2)) as 5x^(-2):
(2)/(y^(2)) = 5x^(-2) + 3
Now, cross-multiply:
2 = (5x^(-2) + 3)y^(2)
Divide both sides by (5x^(-2) + 3) to solve for y^2:
y^(2) = 2 / (5x^(-2) + 3)
Now, substitute this value of y^2 in equation (2):
(3)/(x^(2)) + (1)/(2 / (5x^(-2) + 3)) = 7
Simplify this equation by multiplying both sides by 2(5x^(-2) + 3) to eliminate the fraction:
6(5x^(-2) + 3) + x^(2) = 14(5x^(-2) + 3)
Expand and simplify the equation:
30x^(-2) + 18 + x^(2) = 70x^(-2) + 42
Rearrange the equation:
40x^(-2) - x^(2) = 24
Multiply both sides by x^2 to get rid of the negative exponent:
40 - x^(4) = 24x^(2)
Rearrange the equation:
x^(4) + 24x^(2) - 40 = 0
This is a quadratic equation in terms of x^2. Let's substitute x^2 with a variable, say z:
z^2 + 24z - 40 = 0
Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the possible values of z.
Once you find the values of z, substitute them back into the equation x^2 = z to find the corresponding values of x.
Finally, substitute the values of x into either equation (1) or (2) to find the corresponding values of y.