SOLUTION: {{{ 4/(x^2-5x) -(8/x) = 5/(x-5) }}} X=? I have found that the answer is not "no solution". Thank you SOOO much!! :)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: {{{ 4/(x^2-5x) -(8/x) = 5/(x-5) }}} X=? I have found that the answer is not "no solution". Thank you SOOO much!! :)      Log On


   

Question 709812: +4%2F%28x%5E2-5x%29++-%288%2Fx%29+=+5%2F%28x-5%29+
X=?
I have found that the answer is not "no solution".
Thank you SOOO much!! :)
Found 2 solutions by josmiceli, jsmallt9:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+4%2F%28x%5E2-5x%29++-%288%2Fx%29+=+5%2F%28x-5%29+
+4%2F%28x%2A%28x-5%29%29++-%288%2Fx%29+=+5%2F%28x-5%29+
Multiply both sides by +x%2A%28x-5%29+
+4+-+8%2A%28+x-5+%29+=+5x+
+4+-+8x+%2B+40+=+5x+
+13x+=+44+
+x+=+44%2F13+
--------------
I'll check by converting to decimals
+x+=+3.3846+
+4%2F%283.3846%2A%283.3846-5%29%29++-%288%2F3.3846%29+=+5%2F%283.3846-5%29+
+4+%2F+%28+3.3846%2A%28-1.6154%29%29+-+2.3636+=+5+%2F+%28-1.6154%29+
+4+%2F+%28+-5.4675+%29+-+2.3636+=+-3.0952+
+-.7316+-+2.3636+=+-3.0952+
+-3.0952+=+-3.0952+
OK

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+4%2F%28x%5E2-5x%29++-%288%2Fx%29+=+5%2F%28x-5%29+
The first goal is to eliminate the fractions. Fractions can be eliminated from an equation if both sides of the equation are multiplied by the lowest common denominator (LCD) of all the fractions (on both sides). To find a LCD we should start by factoring each denominator:
x^2-5x  = 1 * x * (x-5)
x       = 1 * x
x-5     = 1 *     (x-5)
LCM/LCD = 1 * x * (x-5)
If you have trouble with LCD's the way I did the factoring above can be helpful. One line per denominator with spacing used for the factors so that each column has the same factors. In this case, one column for the x factors and one column for the (x-5) factors. The 1's are not really needed for LCD's but they are useful when looking for greatest common factors (GCF) so out of habit I include them.) The LCD consists of factors from every column. (The GCF consists of that have an entry in each row. In this case, only the 1 is present in all three lists of factors so the GCF is 1.) NOTE: An LCD is simply the least common multiple (LCM) of some denominators. That is why the bottom line is listed as LCM/LCD.

Now that we have the LCD we know what to multiply each side by:

On the left side we will need to use the Distributive Property to multiply correctly:

In each term, the denominator will cancel with part or all of the LCD. (After all the LCD is a multiple of each denominator!)
4++-+%28x-5%29%288%29+=+%28x%29%285%29+
And the fractions are gone! This simplifies as follows:
4++-+%288x-40%29+=+5x+
4++-+8x%2B40+=+5x+
44++-+8x+=+5x+

Now we solve for x. Adding 8x to each side:
44++=+13x+
And divide by 13:
44%2F13++=+x+

Last of all we check. This is not optional because we multiplied both sides of the equation by an expression, the LCD, which could be zero depending on the value of x. So we must ensure that the solution(s) we found would not make that expression zero. The LCD was:
(x * (x-5))
If x = 44/13:
((44/13) * ((44/13)-5))
We should be able to see that neither factor will be zero. therefore the whole LCD will not be zero. So we did not multiply both sides of the equation by a zero when we multiplied by the LCD. This is the required part of the check.

Checking to see if 44/13 is actually the solution (IOW: to see if we made a mistake) is optional. If you want such a check I'll leave that up to you.