`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**.

**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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