5x+3y=4
5x^2-3y^2=8
You asked:
Solve the system of equations:\[\begin{aligned}5 x + 3 y &= 4\\5 {x}^{2} - 3 {y}^{2} &= 8\end{aligned}\]
MathBot Answer:
The system of equations has one solution.
\[x = 2, y = -2\]
Solve \(5 x + 3 y = 4\) for \(x\). \[x = \frac{4}{5} - \frac{3 y}{5}\]Substitute \(\frac{4}{5} - \frac{3 y}{5}\) for \(x\) in \(5 x^{2} - 3 y^{2} = 8\) and simplify. $$\begin{aligned}5 x^{2} - 3 y^{2} &= 8 \\ 5 \left(\frac{4}{5} - \frac{3 y}{5}\right)^{2} - 3 y^{2} &= 8 \\ y^{2} + 4 y &= -4 \\y^{2} + 4 y + 4 &= 0 \\ \left(y + 2\right)^{2} &= 0 \\ y = -2\end{aligned}$$Substitute \(-2\) into \(5 x + 3 y = 4\) to solve for \(x\). \[\begin{aligned}5 x - 6 &= 4\\5 x &= 10\\x &= 2\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 2,\,y = -2\end{aligned}$$