Math Problem Solver

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

\[x = -9 + \frac{\sqrt{323}}{2}, y = -15 + \sqrt{323}\]\[x = -9 - \frac{\sqrt{323}}{2}, y = - \sqrt{323} - 15\]

Solve \(y = 2 x + 3\) for \(x\). \[x = \frac{y}{2} - \frac{3}{2}\]Substitute \(\frac{y}{2} - \frac{3}{2}\) for \(x\) in \(60 x + y^{2} = 8\) and simplify. $$\begin{aligned}60 x + y^{2} &amp= 8 \\ 60 \left(\frac{y}{2} - \frac{3}{2}\right) + y^{2} &= 8 \\ y^{2} + 30 y &= 98 \\y^{2} + 30 y - 98 &= 0 \\ y &= \frac{-(30) \pm \sqrt{(30)^{2} - 4(1)(-98)}}{2(30)} \\ y = -15 + \sqrt{323}&, y = - \sqrt{323} - 15\end{aligned}$$Substitute \(-15 + \sqrt{323}\) into \(y = 2 x + 3\) to solve for \(x\). \[\begin{aligned}-15 + \sqrt{323} &= 2 x + 3\\- 2 x &= 18 - \sqrt{323}\\x &= -9 + \frac{\sqrt{323}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = -9 + \frac{\sqrt{323}}{2},\,y = -15 + \sqrt{323}\end{aligned}$$Substitute \(- \sqrt{323} - 15\) into \(y = 2 x + 3\) to solve for \(x\). \[\begin{aligned}- \sqrt{323} - 15 &= 2 x + 3\\- 2 x &= \sqrt{323} + 18\\x &= -9 - \frac{\sqrt{323}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = -9 - \frac{\sqrt{323}}{2},\,y = - \sqrt{323} - 15\end{aligned}$$