Q. 33 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The solution of the given system of equation is $x=1.5, y=1; \left(1.5,1\right)$.

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}2x+3y& =& 6\\ x-y& =& \frac{1}{2}\end{array}$

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

The second equation can be written as:

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

Step 3. Determine the value of one variable.

Substitute the above value in the first equation.

$\begin{array}{rcl}2\left(\frac{1}{2}+y\right)+3y& =& 6\\ 1+2y+3y& =& 6\\ 5y& =& 6-1\\ y& =& \frac{5}{5}\\ y& =& 1\end{array}$

Step 4. Determine the value of another variable.

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

$\begin{array}{rcl}x& =& \frac{1}{2}+\left(1\right)\\ x& =& \frac{3}{2}\\ x& =& 1.5\end{array}$

The solution of the given system of equation is $x=1.5, y=1; \left(1.5,1\right)$.