Math Problem Solver

The system of equations has $2$ solutions.

$$x = \frac{39}{2} - \frac{\sqrt{1153}}{2}, y = \frac{\sqrt{1153}}{2} + \frac{39}{2}$$$$x = \frac{\sqrt{1153}}{2} + \frac{39}{2}, y = \frac{39}{2} - \frac{\sqrt{1153}}{2}$$

Solve $x + y = 39$ for $x$. $$x = 39 - y$$Substitute $39 - y$ for $x$ in $x y = 92$ and simplify. $$\begin{aligned}x y &= 92 \\ \left(39 - y\right) y &= 92 \\ y^{2} - 39 y &= -92 \\y^{2} - 39 y + 92 &= 0 \\ y &= \frac{-(-39) \pm \sqrt{(-39)^{2} - 4(1)(92)}}{2(-39)} \\ y = \frac{39}{2} - \frac{\sqrt{1153}}{2}&, y = \frac{\sqrt{1153}}{2} + \frac{39}{2}\end{aligned}$$Substitute $\frac{39}{2} - \frac{\sqrt{1153}}{2}$ into $x + y = 39$ to solve for $x$. $$\begin{aligned}x - \frac{\sqrt{1153}}{2} + \frac{39}{2} &= 39\\x + \left(\frac{39}{2} - \frac{\sqrt{1153}}{2}\right) &= 39\\x &= \frac{\sqrt{1153}}{2} + \frac{39}{2}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{\sqrt{1153}}{2} + \frac{39}{2},\,y = \frac{39}{2} - \frac{\sqrt{1153}}{2}\end{aligned}$$Substitute $\frac{\sqrt{1153}}{2} + \frac{39}{2}$ into $x + y = 39$ to solve for $x$. $$\begin{aligned}x + \frac{\sqrt{1153}}{2} + \frac{39}{2} &= 39\\x + \left(\frac{\sqrt{1153}}{2} + \frac{39}{2}\right) &= 39\\x &= \frac{39}{2} - \frac{\sqrt{1153}}{2}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{39}{2} - \frac{\sqrt{1153}}{2},\,y = \frac{\sqrt{1153}}{2} + \frac{39}{2}\end{aligned}$$