Chapter 9.5, Problem 38WE

To determine

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

Answer to Problem 38WE

The value of $x$ is $\frac{1}{5}$ , and the value of $y$ is $-\frac{1}{4}$ .

Explanation of Solution

Given information:

The given equations are,

  $\begin{array}{l}\frac{1}{x}+\frac{1}{y}=1\\ \frac{3}{x}-\frac{2}{y}=8\end{array}$

Calculation:

Calculate the equations,

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

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

  $\begin{array}{c}\frac{3}{x}-\frac{2}{y}=8\\ 3a-2b=8\end{array}$ … (2)

Multiply by $2$ in equation $\left(1\right)$ .

  $\begin{array}{c}2\left(a+b\right)=2\left(1\right)\\ 2a+2b=2\end{array}$ … (3)

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

  $\begin{array}{c}2a+2b=2\\ 3a-2b=8\end{array}$

  $5a\text{ = 10}$

Solve the result,

  $a=5$

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

  $\begin{array}{c}a+b=1\\ 5+b=1\\ b=1-5\\ b=-4\end{array}$

Therefore,

  $\begin{array}{l}a=5\\ b=-4\end{array}$

Now,

  $\begin{array}{l}\frac{1}{x}=a\to x=\frac{1}{5}\\ \frac{1}{y}=b\to y=-\frac{1}{4}\end{array}$

Therefore, the value of $x$ is $\frac{1}{5}$ , and the value of $y$ is $-\frac{1}{4}$ .