Q. 23 - Precalculus Enhanced with Graphing Utilities

Question

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}3x-6y=2\\ 5x+4y=1\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the given system of equation is $x=\frac{1}{3}, y=-\frac{1}{6}; \left(\frac{1}{3},-\frac{1}{6}\right)$.

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}3x-6y& =& 2\\ 5x+4y& =& 1\end{array}$

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

The first equation can be written as:

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

Step 3. Determine the value of one variable.

Substitute the above value in the second equation.

$\begin{array}{rcl}5\left(\frac{2+6y}{3}\right)+4y& =& 1\\ \frac{10+30y+12y}{3}& =& 1\\ 10+42y& =& 3\\ 42y& =& 3-10\\ y& =& \frac{-7}{42}\\ y& =& -\frac{1}{6}\end{array}$

Step 4. Determine the value of another variable.

Substitute $y=-\frac{1}{6}$ in the first equation.

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

The solution of the given system of equation is $x=\frac{1}{3}, y=-\frac{1}{6}; \left(\frac{1}{3},-\frac{1}{6}\right)$.