Question

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{c}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Short answer

The solution of the given system of equation is $x=\frac{4}{3},y=\frac{1}{5};\left(\frac{4}{3},\frac{1}{5}\right)$.

Step-by-step solution

Step 1. Given information.

The given equations are:

$\begin{array}{rcl}3x-5y& =& 3\\ 15x+5y& =& 21\end{array}$

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

The first equation can be written as:

$\begin{array}{rcl}3x-5y& =& 3\\ 3x& =& 3+5y\\ x& =& \frac{3+5y}{3}\end{array}$

Step 3. Determine the value of one variable.

Substitute the above value in the second equation.

$\begin{array}{rcl}15\left(\frac{3+5y}{3}\right)+5y& =& 21\\ 15+25y+5y& =& 21\\ 30y& =& 21-15\\ y& =& \frac{6}{30}\\ y& =& \frac{1}{5}\end{array}$

Step 4. Determine the value of another variable.

Substitute $y=\frac{1}{5}$in $x=\frac{3+5y}{3}$.

$\begin{array}{rcl}x& =& \frac{3+5\left(\frac{1}{5}\right)}{3}\\ x& =& \frac{3+1}{3}\\ x& =& \frac{4}{3}\end{array}$

The solution of the given system of equation is $x=\frac{4}{3},y=\frac{1}{5};\left(\frac{4}{3},\frac{1}{5}\right)$.