`18/(x^2-3x)-6/(x-3)=5/x` Solve the equation by using the LCD. Check for extraneous solutions.

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`18/(x^2-3x)-6/(x-3)=5/x`

LCD is `x^2-3x=x(x-3)`

Multiply all the terms of the equation by LCD and simplify,

`x(x-3)(18/(x^2-3x))-x(x-3)(6/(x-3))=x(x-3)(5/x)`

`18-6x=(x-3)5`

`18-6x=5x-15`

`-6x-5x=-15-18`

`-11x=-33`

`x=(-33)/(-11)`

`x=3`

Let's check the solution by plugging in the original equation,

`18/(3^2-3*3)-6/(3-3)=5/3`

`18/0-6/0=5/3`

Since the solution x=3, yields a denominator of zero, so it's an extraneous solution and the original equation has no solution.

 

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LCD is an acronym for least common denominator. It is the product of distinct factors on the denominator side. Basically, find LCD is the same as finding the LCM (least common multiple) of the denominators.

For the given equation `18/(x^2-3x)-6/(x-3)=5/x` , the denominators are `(x^2-3x)` , `(x-3)` , and `x` . Note: The factored form of the denominator `x^2-3x` is ` x(x-3)` .

Based on the list of factors, The distinct factors are `x` and `(x-3).`

Thus,` LCD= x*(x-3) `  or `(x^2-3x)` .

To simplify the equation, we multiply each term by the LCD.

`18/(x^2-3x)*(x^2-3x)-6/(x-3)*x*(x-3) =5/x*x*(x-3) `

Cancel out common factors to get rid of the denominators.

`18 -6*x =5*(x-3)`

Apply distribution property.

`18-6x=5x-15`

Subtract 18 from both sides.

`18-6x-18=5x-15-18`

`-6x= 5x -33`

Subtract `5x ` from both sides of the equation.

`-6x-5x= 5x -33-5x`

`-11x=-33`

Divide both sides by `-11` .

`(-11x)/(-11)= (-33)/(-11)`

`x=3`

To check for extraneous solution, plug-in `x=3` on `18/(x^2-3x)-6/(x-3)=5/x` .

`18/(3^2-3*3)-6/(3-3)=?5/3`

`18/(9-9)-6/0=?5/3`

`18/0-6/0=?5/3`

undefined -undefined=?`5/3 `   FALSE.

Note: Any value divided by `0` results to undefined value.

An undefined result  implies the x value is an extraneous solution.

Therefore, the `x=3` is an extraneous solution.

 There is no real solution for the given equation `18/(x^2-3x)-6/(x-3)=5/x` .

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