Question

Solve the system of equations by substitution

$\left\{\begin{array}{l}-2x+2y=6\\ y=-3x+1\end{array}\right.$

Short answer

The solution for the system of equations is,

$$\begin{gathered} x=-\frac{1}{2}, \\ y=\frac{5}{2} \end{gathered}$$

Step-by-step solution

Step 1. Given information

The system of equations is,

$\left\{\begin{array}{l}-2x+2y=6\\ y=-3x+1\end{array}\right.$

Step 2. Substituting the value of y in the equation.

Substitute $y=-3x+1$in the equation

$-2x+2y=6$,

$-2x+2(-3x+1)=6$

$-2x-6x+2=6$

$-8x+2=6$

$-8x=6-2$

$-8x=4$

$x=-\frac{4}{8}$

$x=\frac{-1}{2}$

Step 3. Substituting the value of x in the equation.

Substituting the value of $x=-\frac{1}{2}$in the equation

$y=-3x+1$

$y=-3(-\frac{1}{2})+1$

$y=\frac{3}{2}+1$

$y=\frac{3+2}{2}$

$y=\frac{5}{2}$

Step 4. Checking the solution for the equation y=-3x+1

Substituting $$\begin{gathered} x=-\frac{1}{2} \\ y=\frac{5}{2} \end{gathered}$$in the equation

$y=-3x+1$

$\frac{5}{2}=-3(-\frac{1}{2})+1$

$\frac{5}{2}=\frac{3}{2}+1$

$\frac{5}{2}=\frac{3+2}{2}$

$\frac{5}{2}=\frac{5}{2}$

This is true.

Step 5. Checking the solution for the equation -2x+2y=6.

Substituting $$\begin{gathered} x=-\frac{1}{2}, \\ y=\frac{5}{2} \end{gathered}$$in the equation

$-2x+2y=6$

$-2(-\frac{1}{2})+2\left(\frac{5}{2}\right)=6$

$1+5=6$

$6=6$

This is true.