Chapter 5.9, Problem 24WE

To determine

To solve and check , the given equation has no solution of $\frac{y}{y-2}-\frac{2}{y+3}=\frac{10}{y^2+y-6}$

Answer to Problem 24WE

No solution

Explanation of Solution

Given that: : $\frac{y}{y-2}-\frac{2}{y+3}=\frac{10}{y^2+y-6}$

First factor the $y^2+y-6$

  $\begin{array}{c}{y}^2+y-6=y^2+(3-2)y-6\\ =y^2+3y-2y-6\\ =y(y+3)-2(y+3)\\ =(y+3)(y-2)\end{array}$

Rewrite the given equation

  $\frac{y}{y-2}-\frac{2}{y+3}=\frac{10}{(y-2)(y+3)}$

Multiply by $(y+3)(y-2)$ , LCM of $(y+3)and(y-2)$

  $(y+3)(y-2)\left(\frac{y}{y-2}-\frac{2}{(y+3)}\right)=(y+3)(y-2)\left(\frac{10}{\left(y-2\right)(y+3)}\right)$

Apply distributive property

  $\begin{array}{c}y(y+3)-2(y-2)=10\\ y^2+3y-2y+4=10\\ y^2+y+4=10\\ y^2+y-6=0\end{array}$

Apply the factor method and Apply zero product property to find the solution of equation

  $\begin{array}{c}{y}^2+y-6=y^2+(3-2)y-6\\ =y^2+3y-2y-6\\ =y(y+3)-2(y+3)\\ =(y+3)(y-2)\\ y=-3,2\end{array}$

When    y   equal to -3 and 2, a denominator in the original equation has a value of 0. Therefore,

-3 and 2 are not in the domain of y and cannot be solution.

Factor value exactly same of denominator therefore, the solution is undefined.

Check:

  $\begin{array}{l}Fory=-3\\ \frac{-3}{-3-2}-\frac{2}{-3+3}\ne \frac{10}{9-3-6}\\ Similarly\\ Fory=2\\ \frac{2}{2-2}-\frac{2}{2+3}\ne \frac{10}{4+2-6}\end{array}$

Hence, No solution of equation.