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Note that we can make everything have a common denominator. The common denominator is
formed by multiplying together the denominators of all three fractions and is:
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The first fraction on the left side already has as its denominator. Suppose that
we multiply the first fraction by . Notice that since this
multiplier has the same numerator as its denominator it is the same as multiplying by 1.
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With this multiplication the first fraction becomes:
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Now let's work on the second fraction on the left side.
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We do a similar process on the second fraction which is . This fraction
already has in the denominator. So this fraction we multiply by
because this fraction contains the two factors that are missing for the common denominator.
The result of this multiplication is:
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Finally, let's work on the fraction on the right side of the equal sign. This fraction is:
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and it is missing the two terms and if it is to have the common denominator.
Therefore let's multiply this fraction by . The result is:
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Now we can substitute all 3 of these converted fractions into the original problem to get:
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Now we can multiply all terms on both sides by the common denominator and that will cancel
out the common denominator from all terms just leaving us with the numerators as follows:
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On the right side drop the 1 and just keeping to get:
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Next do all the multiplications. First,
Next,
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Finally do the multiplication on the right side to get:
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Put all these multiplications into the equation to get:
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Combine the like terms on the left side. First you have . Next you have . Finally you have . When you substitute these results
on the left side the entire equation becomes:
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This is further reduced by adding to both sides and this eliminates the squared
terms entirely. What is left is:
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Subtract 12y from both sides to eliminate the 12y on the right side and the equation becomes:
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Get rid of the minus signs by multiplying both sides by -1 and you have:
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and finally divide both sides by 18 to get:
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That's the answer. You can convert it to a proper fraction (1 and 7/9) or a decimal if you
like.
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It's been a long trip to get to the answer, but there is a lot of good practice involved in
getting there. Hope this helps you to gain some insight into algebraic manipulations.
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