Math Problem Solver

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

\[x = -2, y = 8\]\[x = \frac{7}{2}, y = \frac{49}{2}\]

Solve \(y = 2 x^{2}\) for \(x\). \[x = - \frac{\sqrt{2} \sqrt{y}}{2}, x = \frac{\sqrt{2} \sqrt{y}}{2}\]Substitute \(- \frac{\sqrt{2} \sqrt{y}}{2}\) for \(x\) in \(y = 3 x + 14\) and simplify. $$\begin{aligned}y &= 3 x + 14 \\ y &= 3 \left(- \frac{\sqrt{2} \sqrt{y}}{2}\right) + 14 \\ y &= - \frac{3 \sqrt{2} \sqrt{y}}{2} + 14 \end{aligned}$$Substitute \(8\) into \(y = 2 x^{2}\) to solve for \(x\). $$\begin{aligned}8 &= 2 x^{2} \\8 - 2 x^{2} &= 0 \\ - 2 \left(x - 2\right) \left(x + 2\right) &= 0 \\ x = -2&, x = 2\end{aligned}$$This yields the following solution. $$\begin{aligned}x = -2,\,y = 8\end{aligned}$$Substitute \(- \frac{\sqrt{2} \sqrt{y}}{2}\) for \(x\) in \(y = 3 x + 14\) and simplify. $$\begin{aligned}y &= 3 x + 14 \\ y &= 3 \left(- \frac{\sqrt{2} \sqrt{y}}{2}\right) + 14 \\ y &= - \frac{3 \sqrt{2} \sqrt{y}}{2} + 14 \end{aligned}$$Substitute \(\frac{\sqrt{2} \sqrt{y}}{2}\) for \(x\) in \(y = 3 x + 14\) and simplify. $$\begin{aligned}y &= 3 x + 14 \\ y &= 3 \left(\frac{\sqrt{2} \sqrt{y}}{2}\right) + 14 \\ y &= \frac{3 \sqrt{2} \sqrt{y}}{2} + 14 \end{aligned}$$Substitute \(\frac{49}{2}\) into \(y = 2 x^{2}\) to solve for \(x\). $$\begin{aligned}\frac{49}{2} &= 2 x^{2} \\\frac{49}{2} - 2 x^{2} &= 0 \\ - \frac{\left(2 x - 7\right) \left(2 x + 7\right)}{2} &= 0 \\ x = - \frac{7}{2}&, x = \frac{7}{2}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{7}{2},\,y = \frac{49}{2}\end{aligned}$$