Math Problem Solver

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

\[x = - \frac{5}{3}, y = - \frac{80}{3}\]\[x = \frac{5}{2}, y = \frac{5}{2}\]

Solve \(y = - 6 x^{2} + 12 x + 10\) for \(x\). \[x = 1 - \frac{\sqrt{96 - 6 y}}{6}, x = \frac{\sqrt{96 - 6 y}}{6} + 1\]Substitute \(1 - \frac{\sqrt{96 - 6 y}}{6}\) for \(x\) in \(y = 7 x - 15\) and simplify. $$\begin{aligned}y &= 7 x - 15 \\ y &= 7 \left(1 - \frac{\sqrt{96 - 6 y}}{6}\right) - 15 \\ y &= - \frac{7 \sqrt{96 - 6 y}}{6} - 8 \end{aligned}$$Substitute \(- \frac{80}{3}\) into \(y = - 6 x^{2} + 12 x + 10\) to solve for \(x\). $$\begin{aligned}- \frac{80}{3} &= - 6 x^{2} + 12 x + 10 \\6 x^{2} - 12 x - \frac{110}{3} &= 0 \\ \frac{2 \cdot \left(3 x - 11\right) \left(3 x + 5\right)}{3} &= 0 \\ x = - \frac{5}{3}&, x = \frac{11}{3}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = - \frac{5}{3},\,y = - \frac{80}{3}\end{aligned}$$Substitute \(1 - \frac{\sqrt{96 - 6 y}}{6}\) for \(x\) in \(y = 7 x - 15\) and simplify. $$\begin{aligned}y &= 7 x - 15 \\ y &= 7 \left(1 - \frac{\sqrt{96 - 6 y}}{6}\right) - 15 \\ y &= - \frac{7 \sqrt{96 - 6 y}}{6} - 8 \end{aligned}$$Substitute \(\frac{\sqrt{96 - 6 y}}{6} + 1\) for \(x\) in \(y = 7 x - 15\) and simplify. $$\begin{aligned}y &= 7 x - 15 \\ y &= 7 \left(\frac{\sqrt{96 - 6 y}}{6} + 1\right) - 15 \\ y &= \frac{7 \sqrt{96 - 6 y}}{6} - 8 \end{aligned}$$Substitute \(\frac{5}{2}\) into \(y = - 6 x^{2} + 12 x + 10\) to solve for \(x\). $$\begin{aligned}\frac{5}{2} &= - 6 x^{2} + 12 x + 10 \\6 x^{2} - 12 x - \frac{15}{2} &= 0 \\ \frac{3 \cdot \left(2 x - 5\right) \left(2 x + 1\right)}{2} &= 0 \\ x = - \frac{1}{2}&, x = \frac{5}{2}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{5}{2},\,y = \frac{5}{2}\end{aligned}$$