Math Problem Solver

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

\[x = 64 - 3 \sqrt{465}, y = 6 - \frac{\sqrt{465}}{3}\]\[x = 64 + 3 \sqrt{465}, y = 6 + \frac{\sqrt{465}}{3}\]

Solve \(x - 9 y = 10\) for \(x\). \[x = 9 y + 10\]Substitute \(9 y + 10\) for \(x\) in \(3 y^{2} = 4 x + 7\) and simplify. $$\begin{aligned}3 y^{2} &amp= 4 x + 7 \\ 3 y^{2} &= 4 \left(9 y + 10\right) + 7 \\ y^{2} - 12 y &= \frac{47}{3} \\y^{2} - 12 y - \frac{47}{3} &= 0 \\ \frac{3 y^{2} - 36 y - 47}{3} &= 0 \\ y = 6 - \frac{\sqrt{465}}{3}&, y = 6 + \frac{\sqrt{465}}{3}\end{aligned}$$Substitute \(6 - \frac{\sqrt{465}}{3}\) into \(x - 9 y = 10\) to solve for \(x\). \[\begin{aligned}x - 54 + 3 \cdot \sqrt{465} &= 10\\x - \left(54 - 3 \sqrt{465}\right) &= 10\\x &= 64 - 3 \sqrt{465}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 64 - 3 \sqrt{465},\,y = 6 - \frac{\sqrt{465}}{3}\end{aligned}$$Substitute \(6 + \frac{\sqrt{465}}{3}\) into \(x - 9 y = 10\) to solve for \(x\). \[\begin{aligned}x - 3 \cdot \sqrt{465} - 54 &= 10\\x - \left(54 + 3 \sqrt{465}\right) &= 10\\x &= 64 + 3 \sqrt{465}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 64 + 3 \sqrt{465},\,y = 6 + \frac{\sqrt{465}}{3}\end{aligned}$$