Question

In the following exercises, solve the systems of equations by substitution.

$$\begin{gathered} -2x+2y=6 \\ y=-3x+1 \end{gathered}$$

Short answer

The solution set is $\left(-\frac{1}{2},\frac{5}{2}\right)$.

Step-by-step solution

Step 1. Given Information

In the following exercises, we have to solve the systems of equations by substitution.
$$\begin{gathered} -2x+2y=6 \\ y=-3x+1 \end{gathered}$$

Step 2. Putting the value of y from the second equation in the first equation .

$$\begin{gathered} -2x+2(-3x+1)=6 \\ -2x-6x+2=6 \\ -8x+2=6 \\ \mathrm{Subtract}2\mathrm{on}\mathrm{both}\mathrm{side} \\ -8x+2-2=6-2 \\ -8x=4 \\ \mathrm{Divide}\mathrm{by}-8\mathrm{on}\mathrm{both}\mathrm{side} \\ \frac{-8}{-8}x=\frac{4}{-8} \\ x=-\frac{1}{2} \end{gathered}$$

Step 3. Putting the value of x in the equation y=−3x+1.

$$\begin{gathered} y=-3\left(-\frac{1}{2}\right)+1 \\ y=\frac{3}{2}+1 \\ y=\frac{3}{2}+1·\frac{2}{2} \\ y=\frac{3}{2}+\frac{2}{2} \\ y=\frac{3+2}{2} \\ y=\frac{5}{2} \end{gathered}$$