Math Problem Solver

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

\[x = -3, y = -1\]\[x = - \frac{39}{19}, y = \frac{35}{19}\]

Solve \(x^{2} + 2 y^{2} = 11\) for \(x\). \[x = - \sqrt{11 - 2 y^{2}}, x = \sqrt{11 - 2 y^{2}}\]Substitute \(- \sqrt{11 - 2 y^{2}}\) for \(x\) in \(- 3 x + y = 8\) and simplify. $$\begin{aligned}- 3 x + y &amp= 8 \\ - 3 \left(- \sqrt{11 - 2 y^{2}}\right) + y &= 8 \\ y + 3 \sqrt{11 - 2 y^{2}} &= 8 \end{aligned}$$Substitute \(-1\) into \(x^{2} + 2 y^{2} = 11\) to solve for \(x\). $$\begin{aligned}x^{2} + 2 \left(-1\right)^{2} &= 11 \\ x^{2} + 2 &= 11 \\x^{2} - 9 &= 0 \\ \left(x - 3\right) \left(x + 3\right) &= 0 \\ x = -3&, x = 3\end{aligned}$$This yields the following solution. $$\begin{aligned}x = -3,\,y = -1\end{aligned}$$Substitute \(\frac{35}{19}\) into \(x^{2} + 2 y^{2} = 11\) to solve for \(x\). $$\begin{aligned}x^{2} + 2 \left(\frac{35}{19}\right)^{2} &= 11 \\ x^{2} + \frac{2450}{361} &= 11 \\x^{2} - \frac{1521}{361} &= 0 \\ \frac{\left(19 x - 39\right) \left(19 x + 39\right)}{361} &= 0 \\ x = - \frac{39}{19}&, x = \frac{39}{19}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = - \frac{39}{19},\,y = \frac{35}{19}\end{aligned}$$