Question

Solve the following system of equations and show all work.

$$\begin{gathered} y=2x^2 \\ y=-3x-1 \end{gathered}$$

Answer 1

Step-1 : Elimination of the variable $y$

Equate $-3x-1$and $2x^2$ to obtain the value of $x.$

$$\begin{gathered} -3x-1=2x^2 \\ \Rightarrow 2x^2+3x+1=0 \\ \Rightarrow 2x^2+2x+x+1=0 \\ \therefore 2x\left(x+1\right)+x+1=0 \end{gathered}$$

Step-2 : Calculation of the value of $x$:

Simplify the equation $2x\left(x+1\right)+x+1=0$ to calculate the value of $x.$

$$\begin{gathered} 2x(x+1)+x+1=0 \\ \Rightarrow \left(2x+1\right)\left(x+1\right)=0 \\ \therefore x=\frac{-1}{2}\mathrm{or}x=-1 \end{gathered}$$

Step-3 : Calculation of the value of $y$:

Substitute $-1$ and $\frac{-1}{2}$ for $x$ in the equation $y=2x^2$.

$$\begin{gathered} y=2{\left(-1\right)}^2 \\ \therefore y=2 \\ y=2{\left(\frac{-1}{2}\right)}^2 \\ \therefore y=\frac{1}{2} \end{gathered}$$

Hence, the solution of the given equation is $\left(-1,2\right)$ and $\left(\frac{-1}{2},\frac{1}{2}\right)$.