Q. 38 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The solution of the given system of equation is $x=\frac{1}{2}, y=2; \left(\frac{1}{2},2\right)$.

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}2x-y& =& -1\\ x+\frac{1}{2}y& =& \frac{3}{2}\end{array}$

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

The first equation can be written as:

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

Step 3. Determine the value of one variable.

Substitute the above value in the second equation.

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

Step 4. Determine the value of another variable.

Substitute $x=\frac{1}{2}$ in $y=1+2x$.

$\begin{array}{rcl}y& =& 1+2\left(\frac{1}{2}\right)\\ y& =& 1+1\\ y& =& 2\end{array}$

The solution of the given system of equation is $x=\frac{1}{2}, y=2; \left(\frac{1}{2},2\right)$.