Chapter 9.5, Problem 39WE

To determine

To find: The value of $x\text{ and y}$ by any method.

Answer to Problem 39WE

The value of $x$ is $\frac{1}{2}$ , and the value of $y$ is $\frac{1}{3}$ .

Explanation of Solution

Given information:

The given equations are,

  $\begin{array}{l}\frac{1}{x}+\frac{1}{y}=5\\ \frac{3}{x}-\frac{5}{y}=-9\end{array}$

Calculation:

Calculate the equations,

Substitute $\frac{1}{x}=a$ and $\frac{1}{y}=b$ .

  $\begin{array}{c}\frac{1}{x}+\frac{1}{y}=5\\ a+b=5\end{array}$ … (1)

  $\begin{array}{c}\frac{3}{x}-\frac{5}{y}=-9\\ 3a-5b=-9\end{array}$ … (2)

Multiply by $5$ in equation $\left(1\right)$ .

  $\begin{array}{c}5\left(a+b\right)=5\left(5\right)\\ 5a+5b=25\end{array}$ … (3)

Add equation $\left(2\right)$ and equation $\left(3\right)$ .

  $\begin{array}{c}3a-5b=-9\\ 5a+5b=25\end{array}$

  $8a\text{ = 16}$

Solve the result,

  $a=2$

Substitute the value of $a$ in either equation to get $b$ .

  $\begin{array}{c}a+b=5\\ 2+b=5\\ b=3\end{array}$

Therefore,

  $\begin{array}{l}a=2\\ b=3\end{array}$

Now,

  $\begin{array}{l}\frac{1}{x}=a\to x=\frac{1}{2}\\ \frac{1}{y}=b\to y=\frac{1}{3}\end{array}$

Therefore, the value of $x$ is $\frac{1}{2}$ , and the value of $y$ is $\frac{1}{3}$ .