Math Problem Solver

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

\[x = 2 - 3 \sqrt{5}, y = 3 + 2 \sqrt{5}\]\[x = 2 + 3 \sqrt{5}, y = 3 - 2 \sqrt{5}\]

Solve \(2 x + 3 y = 13\) for \(x\). \[x = \frac{13}{2} - \frac{3 y}{2}\]Substitute \(\frac{13}{2} - \frac{3 y}{2}\) for \(x\) in \(x^{2} + y^{2} = 78\) and simplify. $$\begin{aligned}x^{2} + y^{2} &= 78 \\ \left(\frac{13}{2} - \frac{3 y}{2}\right)^{2} + y^{2} &= 78 \\ y^{2} - 6 y &= 11 \\y^{2} - 6 y - 11 &= 0 \\ y &= \frac{-(-6) \pm \sqrt{(-6)^{2} - 4(1)(-11)}}{2(-6)} \\ y = 3 - 2 \sqrt{5}&, y = 3 + 2 \sqrt{5}\end{aligned}$$Substitute \(3 - 2 \sqrt{5}\) into \(2 x + 3 y = 13\) to solve for \(x\). \[\begin{aligned}2 x - 6 \cdot \sqrt{5} + 9 &= 13\\2 x + \left(9 - 6 \sqrt{5}\right) &= 13\\2 x &= 4 + 6 \sqrt{5}\\x &= 2 + 3 \sqrt{5}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 2 + 3 \sqrt{5},\,y = 3 - 2 \sqrt{5}\end{aligned}$$Substitute \(3 + 2 \sqrt{5}\) into \(2 x + 3 y = 13\) to solve for \(x\). \[\begin{aligned}2 x + 9 + 6 \cdot \sqrt{5} &= 13\\2 x + \left(9 + 6 \sqrt{5}\right) &= 13\\2 x &= 4 - 6 \sqrt{5}\\x &= 2 - 3 \sqrt{5}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 2 - 3 \sqrt{5},\,y = 3 + 2 \sqrt{5}\end{aligned}$$