Solve the system by the substitution method. \[ \begin{aligned} x y & =3 \\ x^{2}+y^{2} & =10 \end{aligned} \]

Step 1 ：We are given the system of equations: \(xy = 3\) and \(x^{2} + y^{2} = 10\)

Step 2 ：First, we rearrange the first equation to solve for one variable in terms of the other. We get \(x = \frac{3}{y}\)

Step 3 ：Next, we substitute this expression into the second equation to solve for the other variable. This gives us \(\left(\frac{3}{y}\right)^{2} + y^{2} = 10\), which simplifies to \(y^{2} + \frac{9}{y^{2}} = 10\)

Step 4 ：Solving this equation gives us four possible values for y: -3, -1, 1, 3

Step 5 ：We then substitute these values back into the first equation to find the corresponding values for x. This gives us four possible pairs of solutions: (-1, -3), (-3, -1), (1, 3), and (3, 1)

Step 6 ：\(\boxed{\text{Final Answer: The solutions to the system of equations are } (-1, -3), (-3, -1), (1, 3), \text{ and } (3, 1)}\)