Question

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{c}\frac{1}{2}x+y=-2\\ x-2y=8\end{array}\right.$

Short answer

The solution of the given system of equation is $x=2,y=-3;\left(2,-3\right)$.

Step-by-step solution

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}\frac{1}{2}x+y& =& -2\\ x-2y& =& 8\end{array}$

Step 2. Determine one variable in terms of another variable.

The second equation can be written as:

$\begin{array}{rcl}x-2y& =& 8\\ x& =& 8+2y\end{array}$

Step 3. Determine the value of one variable.

Substitute the above value in the first equation.

$\begin{array}{rcl}\frac{1}{2}(8+2y)+y& =& -2\\ 4+y+y& =& -2\\ 2y& =& -2-4\\ y& =& \frac{-6}{2}\\ y& =& -3\end{array}$

Step 4. Determine the value of another variable.

Substitute $y=-3$in $x=8+2y$.

$$\begin{gathered} x=8+2y \\ x=8+2(-3) \\ x=8-6 \\ x=2 \end{gathered}$$

The solution of the given system of equation is $x=2,y=-3;\left(2,-3\right)$.