Chapter 3.5, Problem 38WE

To determine

To solve:the given system of the equation.

Answer to Problem 38WE

The solution of the given system is $\left(\frac{2}{3},-1\right)$ .

Explanation of Solution

Given information:

The system of the equation is $\frac{4}{u}+\frac{3}{v}=3\text{ and}\frac{6}{u}+\frac{5}{v}=4$ .

Calculation:

It is given that the variables are u and v.

The equations $\frac{4}{u}+\frac{3}{v}=3\text{ and}\frac{6}{u}+\frac{5}{v}=4$ are not linear with respect to the variable u and v .

The equations can be written in the linear form by give the suitable substitutions.

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

Substitute $x=\frac{1}{u}\text{ and }y=\frac{1}{v}$ in $\frac{4}{u}+\frac{3}{v}=3\text{ and}\frac{6}{u}+\frac{5}{v}=4$ , the equations are $4x+3y=3$ and $6x+5y=4$ .

Compute the solutions of the system $4x+3y=3$ and $6x+5y=4$ by linear combination method.

Find the linear combination that eliminates the variable x .

Since the least common multiple of the x’s coefficients 4 and 6 is 12, multiply the first equation by $-3$ and multiply the second equation by 2,

  $\begin{array}{l}-12x-9y=-9\\ 12x+6y=6\end{array}$

Add these two new equations to eliminate the variable x .

  $\begin{array}{l}-12x-9y=-9\\ 12x+6y=6\\ \overline{\text{ 0}x-3y=-3}\end{array}$

Simplify further to find the value of y .

  $\begin{array}{l}-3y=-3\\ y=-1\end{array}$

Substitute $y=-1$ in $4x+3y=3$ and simplify the value of x ,

  $\begin{array}{l}4x+3\left(-1\right)=3\\ 4x-3=3\\ 4x=6\\ x=\frac{6}{4}\left[\because \text{ Divide on both sides by 4}\right]\\ x=\frac{3}{2}\end{array}$

Thus, the value of x and y are $x=\frac{3}{2}$ and $y=-1$ , respectively.

Substitute $x=\frac{3}{2}$ in $x=\frac{1}{u}$ to find the original variable u ,

  $\begin{array}{l}x=\frac{1}{u}\\ u=\frac{1}{x}\\ u=\frac{1}{\frac{3}{2}}\\ u=\frac{2}{3}\end{array}$

Substitute $y=-1$ in $y=\frac{1}{v}$ to find the original variable v ,

  $\begin{array}{l}v=\frac{1}{y}\\ v=\frac{1}{-1}\\ v=-1\end{array}$

Therefore, the solution of the given system is $\left(\frac{2}{3},-1\right)$ .