Equations involving algebraic fractions

Equations involving algebraic fractions

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The techniques that we have already discussed will now be put in place to help us solve equations that use algebraic fractions. As long as we use the techniques correctly and make sure to do the same multiplications and divisions to both the top and bottom fractions then we should not have any troubles.

Combining all of the techniques we have seen will enable us to break up quite complicated problems into smaller parts that are easier to work out.

Example

Find the value of x in the equation:

\frac{2}{(x+2)}-\frac{1}{(x-3)}=2

By putting the fractions on the left-hand side into ones which have a common denominator we get:

\frac{2(x-3)}{(x+2)(x-3)}-\frac{1(x+2)}{(x-3)(x+2)}=2

Then we can work out the subtraction involved and put the entire left over the same denominator:

\frac{2(x-3)-1(x+2)}{(x+2)(x-3)}=2

Multiplying out the brackets at the top and collecting any like terms, we end up with:

\frac{x-8}{(x+2)(x-3)}=2

Next, we can multiply both sides by the denominator which should leave us with an equation similar to ones which we have already seen in a previous lesson:

x-8=2(x+2)(x-3)=2x^2-2x-12

Now moving all of the terms to the same side we are left with an equation that can be solved by methods of factorisation or using the quadratic formula.

2x^2-3x-4=0

The resulting equation can then be solved using the quadratic formula which we introduced in an earlier lesson to give the answer of x=2.35 or -0.85.

As long as the techniques that we have learnt are used (and a little common sense as to which is the best to use first) then you should not go wrong on these types of questions. Obviously, mistakes may be made so it is always a great idea to check your answer after you have found it.

Checking an answer

Having found an answer to a question we should always double check it, as when a few different techniques are all used in the same equation, it can be easy to make a little mistake which will cause the answer to be incorrect. Fortunately, it is rather easy to do this for an equation, we just put the values of the unknowns which we found back into the original equation that we were given. This then should make the equation correct.

Example

For the previous example we got the answers of 2.35 and -0.85 which need to be put back into the left-hand side of:

\frac{2}{(x+2)}-\frac{1}{(x-3)}=2

This gives an answer of 1.99 and 1.99, which are incredibly close to the true answer of 2 and only different due to the fact that we rounded our answers to 2dp. Therefore, our answers are correct!

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