Chapter 12.3, Problem 26WE

To determine

To solve: the equation $\frac{1}{3}-\frac{2}{2y+1}=\frac{3}{y}$

Answer to Problem 26WE

The required answer is: solution set is - 0.62, - 3.64

Explanation of Solution

Given information:

The given equation: $\frac{1}{3}-\frac{2}{2y+1}=\frac{3}{y}$

Calculation:

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

(Subtract $\frac{1}{3}$ from both sides)

  $\begin{array}{l}-\frac{2y}{2y+1}=\frac{3}{y}-\frac{1}{3}\\ -(\frac{2y}{2y+1})\cdot 2y+1=(\frac{3}{y}-\frac{1}{3})\cdot 2y+1\end{array}$

(Multiply $2y+1$ on both sides)

  $\begin{array}{l}-2y=6+\frac{3}{y}-\frac{2y}{3}-\frac{1}{3}\\ -2y+2y=6+\frac{3}{y}-\frac{2y}{3}-\frac{1}{3}+2y\end{array}$

(Add 2y to both sides)

  $\begin{array}{l}0=6+\frac{3}{y}-\frac{2y}{3}-\frac{1}{3}+2y\\ 0-\frac{3}{y}=6+\frac{3}{y}-\frac{2y}{3}-\frac{1}{3}+2y-\frac{3}{y}\end{array}$

(Subtract $\frac{3}{y}$ from both sides)

  $\begin{array}{l}-\frac{3}{y}=6-\frac{2y}{3}-\frac{1}{3}+2y\\ -\frac{3}{y}\cdot y=(6-\frac{2y}{3}-\frac{1}{3}+2y)\cdot y\end{array}$

(Multiply y on both sides)

  $\begin{array}{l}-3=6y-\frac{2y^2}{3}-\frac{1y}{3}+2y^2\\ -3+3=6y-\frac{2y^2}{3}-\frac{1y}{3}+2y^2+3\end{array}$

(Add 3 to both sides)

  $\begin{array}{l}0=6y-\frac{2y^2}{3}-\frac{1y}{3}+2y^2+3\\ 0=6y-0.67y^2-0.34y+2y^2+3\end{array}$

(Convert fractions to decimals)

  $\begin{array}{l}0=-0.67y^2+2y^2-0.34y+6y+3\\ 0=1.33y^2+5.66y+3\\ 1.33y^2+5.66y+3=0\end{array}$

(Simplify)

  $y=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

(Quadratic formula)

Where $a=1.33,b=5.66andc=3$

  $y=\frac{-(5.66)\pm \sqrt{{(5.66)}^2-4(1.33)(3)}}{2(1.33)}$

(Remove the brackets)

  $\begin{array}{l}=\frac{-5.66\pm \sqrt{32.04-15.96}}{2.66}\\ =\frac{-5.66\pm \sqrt{16.08}}{2.66}\end{array}$

(Simplify)

Since $\sqrt{16.08}\approx 4.01$ , $y\approx \frac{-5.66+4.01}{2.66}=\frac{-1.65}{2.66}\approx -0.62$

Or   $y\approx \frac{-5.66-4.01}{2.66}=\frac{-9.67}{2.66}\approx -3.64$

Therefore the solution set is $\frac{-5.66+\sqrt{16.08}}{2.66},\frac{-5.66-\sqrt{16.08}}{2.66}$ or - 0.62, - 3.64