y=-2x-5,-3x^2+4x-y=8
You asked:
Solve the system of equations:\[\begin{aligned}y &= -\left( 2 x \right) - 5\\-\left( 3 \cdot {x}^{2} \right) + 4 x - y &= 8\end{aligned}\]
MathBot Answer:
The system of equations has one solution.
\[x = 1, y = -7\]
Solve \(y = - 2 x - 5\) for \(x\). \[x = - \frac{y}{2} - \frac{5}{2}\]Substitute \(- \frac{y}{2} - \frac{5}{2}\) for \(x\) in \(- 3 x^{2} + 4 x - y = 8\) and simplify. $$\begin{aligned}- 3 x^{2} + 4 x - y &= 8 \\ - 3 \left(- \frac{y}{2} - \frac{5}{2}\right)^{2} + 4 \left(- \frac{y}{2} - \frac{5}{2}\right) - y &= 8 \\ y^{2} + 14 y &= -49 \\y^{2} + 14 y + 49 &= 0 \\ \left(y + 7\right)^{2} &= 0 \\ y = -7\end{aligned}$$Substitute \(-7\) into \(y = - 2 x - 5\) to solve for \(x\). \[\begin{aligned}-7 &= - 2 x - 5\\2 x &= 2\\x &= 1\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 1,\,y = -7\end{aligned}$$