Math Problem Solver

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

\[x = \frac{2}{5}, y = \frac{14}{5}\]\[x = 2, y = 2\]

Solve \(x + 2 y = 6\) for \(x\). \[x = 6 - 2 y\]Substitute \(6 - 2 y\) for \(x\) in \(x^{2} + y^{2} = 8\) and simplify. $$\begin{aligned}x^{2} + y^{2} &= 8 \\ \left(6 - 2 y\right)^{2} + y^{2} &= 8 \\ 5 y^{2} - 24 y &= -28 \\5 y^{2} - 24 y + 28 &= 0 \\ \left(y - 2\right) \left(5 y - 14\right) &= 0 \\ y = 2&, y = \frac{14}{5}\end{aligned}$$Substitute \(2\) into \(x + 2 y = 6\) to solve for \(x\). \[\begin{aligned}x + 4 &= 6\\x &= 2\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 2,\,y = 2\end{aligned}$$Substitute \(\frac{14}{5}\) into \(x + 2 y = 6\) to solve for \(x\). \[\begin{aligned}x + \frac{28}{5} &= 6\\x &= \frac{2}{5}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{2}{5},\,y = \frac{14}{5}\end{aligned}$$