Math Problem Solver

The system of equations has \(2\) solutions.

\[x = -1, y = -3\]\[x = 3, y = 1\]

Solve \(x^{2} + y^{2} = 10\) for \(x\). \[x = - \sqrt{10 - y^{2}}, x = \sqrt{10 - y^{2}}\]Substitute \(- \sqrt{10 - y^{2}}\) for \(x\) in \(x - y = 2\) and simplify. $$\begin{aligned}x - y &amp= 2 \\ \left(- \sqrt{10 - y^{2}}\right) - y &= 2 \\ y + \sqrt{10 - y^{2}} &= -2 \end{aligned}$$Substitute \(-3\) into \(x^{2} + y^{2} = 10\) to solve for \(x\). $$\begin{aligned}x^{2} + \left(-3\right)^{2} &= 10 \\ x^{2} + 9 &= 10 \\x^{2} - 1 &= 0 \\ \left(x - 1\right) \left(x + 1\right) &= 0 \\ x = -1&, x = 1\end{aligned}$$This yields the following solution. $$\begin{aligned}x = -1,\,y = -3\end{aligned}$$Substitute \(- \sqrt{10 - y^{2}}\) for \(x\) in \(x - y = 2\) and simplify. $$\begin{aligned}x - y &amp= 2 \\ \left(- \sqrt{10 - y^{2}}\right) - y &= 2 \\ y + \sqrt{10 - y^{2}} &= -2 \end{aligned}$$Substitute \(\sqrt{10 - y^{2}}\) for \(x\) in \(x - y = 2\) and simplify. $$\begin{aligned}x - y &amp= 2 \\ \left(\sqrt{10 - y^{2}}\right) - y &= 2 \\ y - \sqrt{10 - y^{2}} &= -2 \end{aligned}$$Substitute \(1\) into \(x^{2} + y^{2} = 10\) to solve for \(x\). $$\begin{aligned}x^{2} + 1^{2} &= 10 \\ x^{2} + 1 &= 10 \\x^{2} - 9 &= 0 \\ \left(x - 3\right) \left(x + 3\right) &= 0 \\ x = -3&, x = 3\end{aligned}$$This yields the following solution. $$\begin{aligned}x = 3,\,y = 1\end{aligned}$$