Math Problem Solver

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

\[x = \frac{11}{5}, y = - \frac{23}{5}\]\[x = \frac{19}{5}, y = - \frac{17}{5}\]

Solve \(3 x - 4 y = 25\) for \(x\). \[x = \frac{4 y}{3} + \frac{25}{3}\]Substitute \(\frac{4 y}{3} + \frac{25}{3}\) for \(x\) in \(x^{2} + y^{2} = 26\) and simplify. $$\begin{aligned}x^{2} + y^{2} &amp= 26 \\ \left(\frac{4 y}{3} + \frac{25}{3}\right)^{2} + y^{2} &= 26 \\ y^{2} + 8 y &= - \frac{391}{25} \\y^{2} + 8 y + \frac{391}{25} &= 0 \\ \frac{\left(5 y + 17\right) \left(5 y + 23\right)}{25} &= 0 \\ y = - \frac{23}{5}&, y = - \frac{17}{5}\end{aligned}$$Substitute \(- \frac{23}{5}\) into \(3 x - 4 y = 25\) to solve for \(x\). \[\begin{aligned}3 x + \frac{92}{5} &= 25\\3 x &= \frac{33}{5}\\x &= \frac{11}{5}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{11}{5},\,y = - \frac{23}{5}\end{aligned}$$Substitute \(- \frac{17}{5}\) into \(3 x - 4 y = 25\) to solve for \(x\). \[\begin{aligned}3 x + \frac{68}{5} &= 25\\3 x &= \frac{57}{5}\\x &= \frac{19}{5}\end{aligned}\]This yields the following solution. $$\begin{aligned}x = \frac{19}{5},\,y = - \frac{17}{5}\end{aligned}$$