Solve the system of equations.

$\begin{array}{c}\frac{1}{x}+\frac{3}{y}=\frac{3}{4}\\ \frac{3}{x}-\frac{2}{y}=\frac{5}{12}\end{array}$

The algebraic method of elimination involves adding or subtracting the equations to eliminate one of the variables and forming new equation that is true. Sometimes, direct addition or subtraction of equations does not eliminate the variable then one equation requires formation of equivalent equation through multiplication so that one of the two variables has the same or opposite coefficient in both the equations. Multiplying the equation by a nonzero number, resulting new equation has same set of solutions.

First, let us assumethat $\frac{1}{x}$is mand $\frac{1}{y}$is n.Rewriting the given equations by substituting the values of $\frac{1}{x}$and $\frac{1}{y}$as.

$\begin{array}{c}m+3n=\frac{3}{4}\\ 3m-2n=\frac{5}{12}\end{array}$

To solve the equations, multiply $m+3n=\frac{3}{4}$by $8$and $3m-2n=\frac{5}{12}$by $12$then add the resulting equations as shown below.

localid="1647437556758" $\begin{array}{c}\left(8\right)\left(m+3n\right)=\left(8\right)\left(\frac{3}{4}\right)\\ 8m+24n=6\end{array}$

$\begin{array}{c}\left(12\right)\left(3m-2n\right)=\left(12\right)\frac{5}{12}\\ 36m-24n=5\end{array}$

Now, add $8m+24=6$and the equation $36m-24n=5$ and solve.

$\begin{array}{ccccccc}& 8m& +& 24n& =& & 6\\ & 36m& -& 24n& =& & 5\\ & & & & & & \\ & 44m& +& 0& =& & 11\end{array}$

Simplify it further as

$\begin{array}{c}44m=11\\ m=\frac{11}{44}\\ m=\frac{1}{4}\end{array}$

Thus, the value of m is $\frac{1}{4}$.

To find the value of n, substitute $m=\frac{1}{4}$in the equation $4m+12n=3$and then solve as shown.

$\begin{array}{c}4m+12n=3\\ 4\left(\frac{1}{4}\right)+12n=3\\ 1+12n=3\\ 12n=2\end{array}$

Simplify it further as shown.

$\begin{array}{c}12n=2\\ n=\frac{2}{12}\\ n=\frac{1}{6}\end{array}$

Thus, the value of n is $\frac{1}{6}$.

Now, substitute the values of mand n in place of $\frac{1}{x}$and $\frac{1}{y}$ to find the values of xand y.

$\begin{array}{c}\frac{1}{x}=m\\ \frac{1}{x}=\frac{1}{4}\\ x=4\end{array}$

$\begin{array}{c}\frac{1}{y}=n\\ \frac{1}{y}=\frac{1}{6}\\ y=6\end{array}$

Hence, the solution of the provided system of equations is $\left(4 , 6\right)$.