Math Problem Solver

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

\[x = \frac{21}{2} - \frac{\sqrt{95}}{2}, y = \frac{\sqrt{95}}{2} + \frac{21}{2}\]\[x = \frac{\sqrt{95}}{2} + \frac{21}{2}, y = \frac{21}{2} - \frac{\sqrt{95}}{2}\]

Solve \(x + y = 21\) for \(x\). \[x = 21 - y\]Substitute \(21 - y\) for \(x\) in \(x^{2} + y^{2} = 268\) and simplify. $$\begin{aligned}x^{2} + y^{2} &= 268 \\ \left(21 - y\right)^{2} + y^{2} &= 268 \\ y^{2} - 21 y &= - \frac{173}{2} \\y^{2} - 21 y + \frac{173}{2} &= 0 \\ \frac{2 y^{2} - 42 y + 173}{2} &= 0 \\ y = \frac{21}{2} - \frac{\sqrt{95}}{2}&, y = \frac{\sqrt{95}}{2} + \frac{21}{2}\end{aligned}$$Substitute \(\frac{21}{2} - \frac{\sqrt{95}}{2}\) into \(x + y = 21\) to solve for \(x\). \[\begin{aligned}x - \frac{\sqrt{95}}{2} + \frac{21}{2} &= 21\\x + \left(\frac{21}{2} - \frac{\sqrt{95}}{2}\right) &= 21\\x &= \frac{\sqrt{95}}{2} + \frac{21}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{\sqrt{95}}{2} + \frac{21}{2},\,y = \frac{21}{2} - \frac{\sqrt{95}}{2}\end{aligned}$$Substitute \(\frac{\sqrt{95}}{2} + \frac{21}{2}\) into \(x + y = 21\) to solve for \(x\). \[\begin{aligned}x + \frac{\sqrt{95}}{2} + \frac{21}{2} &= 21\\x + \left(\frac{\sqrt{95}}{2} + \frac{21}{2}\right) &= 21\\x &= \frac{21}{2} - \frac{\sqrt{95}}{2}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{21}{2} - \frac{\sqrt{95}}{2},\,y = \frac{\sqrt{95}}{2} + \frac{21}{2}\end{aligned}$$