Math Problem Solver

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

\[x = - \frac{5}{2} + \frac{\sqrt{47}}{2}, y = \frac{5}{2} + \frac{\sqrt{47}}{2}\]\[x = - \frac{\sqrt{47}}{2} - \frac{5}{2}, y = \frac{5}{2} - \frac{\sqrt{47}}{2}\]

Solve \(y = x + 5\) for \(x\). \[x = y - 5\]Substitute \(y - 5\) for \(x\) in \(x^{2} + y^{2} = 36\) and simplify. $$\begin{aligned}x^{2} + y^{2} &amp= 36 \\ \left(y - 5\right)^{2} + y^{2} &= 36 \\ y^{2} - 5 y &= \frac{11}{2} \\y^{2} - 5 y - \frac{11}{2} &= 0 \\ \frac{2 y^{2} - 10 y - 11}{2} &= 0 \\ y = \frac{5}{2} - \frac{\sqrt{47}}{2}&, y = \frac{5}{2} + \frac{\sqrt{47}}{2}\end{aligned}$$Substitute \(\frac{5}{2} - \frac{\sqrt{47}}{2}\) into \(y = x + 5\) to solve for \(x\). \[\begin{aligned}\frac{5}{2} - \frac{\sqrt{47}}{2} &= x + 5\\- x &= \frac{5}{2} + \frac{\sqrt{47}}{2}\\x &= - \frac{\sqrt{47}}{2} - \frac{5}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = - \frac{\sqrt{47}}{2} - \frac{5}{2},\,y = \frac{5}{2} - \frac{\sqrt{47}}{2}\end{aligned}$$Substitute \(\frac{5}{2} + \frac{\sqrt{47}}{2}\) into \(y = x + 5\) to solve for \(x\). \[\begin{aligned}\frac{5}{2} + \frac{\sqrt{47}}{2} &= x + 5\\- x &= \frac{5}{2} - \frac{\sqrt{47}}{2}\\x &= - \frac{5}{2} + \frac{\sqrt{47}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = - \frac{5}{2} + \frac{\sqrt{47}}{2},\,y = \frac{5}{2} + \frac{\sqrt{47}}{2}\end{aligned}$$