x^2 + y^2 &= 10 \\xy &= 6\end{cases} \]

This is a system of two equations. Let's solve it step by step.

1. We have two equations here:

   • x^2 + y^2 = 10
   • xy = 6

2. From the second equation, we can express y in terms of x:

   • y = 6/x

3. Now, we can substitute y in the first equation with 6/x:

   • x^2 + (6/x)^2 = 10
   • x^2 + 36/x^2 = 10
   • Multiply through by x^2 to clear the denominator:
   • x^4 + 36 = 10x^2
   • Rearrange to form a quadratic equation:
   • x^4 - 10x^2 + 36 = 0

4. Let's make a substitution to solve this quadratic equation. Let u = x^2. Then the equation becomes:

   • u^2 - 10u + 36 = 0

5. This can be factored into:

   • (u - 6)(u - 4) = 0

6. Setting each factor equal to zero gives the solutions u = 6 and u = 4.

7. But remember, u = x^2. So x^2 = 6 and x^2 = 4. Therefore, the solutions for x are x = sqrt(6), -sqrt(6), 2, -2.

8. Substitute x back into the equation y = 6/x to find the corresponding y values. For x = sqrt(6), y = 6/sqrt(6) = sqrt(6). For x = -sqrt(6), y = -sqrt(6). For x = 2, y = 3. For x = -2, y = -3.

So the solutions to the system of equations are (sqrt(6), sqrt(6)), (-sqrt(6), -sqrt(6)), (2, 3), and (-2, -3).