Q. 39 - Precalculus Enhanced with Graphing Utilities

Question

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

$\left\{\begin{array}{c}\frac{1}{x}+\frac{1}{y}=8\\ \frac{3}{x}-\frac{5}{y}=0\end{array}\right.$

Step-by-Step Solution

Answer

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

Step 1. Given information.

The given equations are:

$$\begin{gathered} \frac{1}{x}+\frac{1}{y}=8 \\ \frac{3}{x}-\frac{5}{y}=0 \end{gathered}$$

Step 2. Rewrite the equations.

Let $u=\frac{1}{x}$ and $v=\frac{1}{y}$. Then, $x=\frac{1}{u}$ and $y=\frac{1}{v}$.

The given equations can be written as:

$\begin{array}{rcl}u+v& =& 8 ...\left(i\right)\\ 3u-5v& =& 0 ...\left(ii\right)\end{array}$

Equation (i) can be written as:

$u=8-v ...\left(iii\right)$

Step 3. Determine the value of one variable.

Substitute the above value in equation (ii)

$\begin{array}{rcl}3(8-v)-5v& =& 0\\ 24-3v-5v& =& 0\\ -8v& =& -24\\ v& =& \frac{-24}{-8}\\ v& =& 3\end{array}$

Step 4. Determine the value of another variable.

Substitute $v=3$ in equation (iii).

$$\begin{gathered} u=8-3 \\ u=5 \end{gathered}$$

We know that $x=\frac{1}{u}$ and $y=\frac{1}{v}$. So, $x=\frac{1}{5}$ and $y=\frac{1}{3}$.

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