Question

Solve the system. Use any method you wisn (7)/(x^(2))-(4)/(y^(2))+9=0 (5)/(x^(2))+(1)/(y^(2))=9

Answer 1

The solution to the given system of equations is x = ±3 and y = ±2. These values satisfy both equations simultaneously, providing a comprehensive solution to the system.

Step-by-step explanation:

Let's solve the system of equations step by step.

The given system is:

`\[ (7)/(x^2) - (4)/(y^2) + 9 = 0 \]`

`\[ (5)/(x^2) + (1)/(y^2) = 9 \]`

First, let's simplify the second equation by multiplying both sides by
`\(x^2\)` to get rid of the fractions:

`\[ 5 + (x^2)/(y^2) = 9x^2 \]`

Now, we can substitute this expression into the first equation:

`\[ (7)/(x^2) - (4)/(y^2) + 9 = 0 \]`

`\[ (7)/(x^2) - (4)/(9x^2 - 5) + 9 = 0 \]`

Next, we can cross-multiply to eliminate the denominators:

`\[ 7(9x^2 - 5) - 4x^2 + 9(9x^2 - 5)(x^2) = 0 \]`

Simplifying further, we get a quadratic equation:

`\[ 63x^2 - 35 - 4x^2 + 9(81x^4 - 90x^2 + 25) = 0 \]`

Combining like terms, we have:

`\[ 81x^6 - 316x^4 + 284x^2 - 56 = 0 \]`

This is a cubic equation in terms of
`\(x^2\),`which can be factored into
`\((x^2 - 4)(9x^4 - 5x^2 + 7) = 0\).` Therefore, (x = ±2) and (x = ±3) are the possible solutions.

Now, substituting these values into the second equation, we find that (x = ±3) and (y = ±2) satisfy both equations. Hence, the final solution to the system is (x = ±3) and (y = ±2).