```Question 91998
I've looked at the two answers posted, and I think they both have mistakes. I think the
first answer interpreted the problem incorrectly for the reason that I will discuss at
the end of my work. And, although Stan H read between the lines and interpreted the problem
the way you meant it to be, he also made a couple of mistakes ... one in multiplying
2(x + 7) and another one in forgetting that the 4 on the right side of the equation has
a minus sign.
You can compare my answer (closer to Stan H's than the other party) and maybe come up with
your own version of what is correct.  Anyhow, here goes:
.
I think the problem you intended to write is:
.
{{{y[1]= 1/(x+7)}}}
{{{y[2]= 2/(x+3)}}} and
{{{y[3]= -4/(x^2 +10x +21)}}}
,
With the added condition that {{{y[1]+y[2]= y[3]}}}
.
Let’s go right to the “added condition” equation and substitute for {{{y[1]}}}, for {{{y[2]}}},
and for {{{y[3]}}} to get:
.
{{{1/(x+7) + 2/(x+3) = -4/(x^2 +10x+21)}}}
.
Notice that on the right side, the denominator can be factored to {{{(x+7)*(x+3)}}}.
This makes the equation become:
.
{{{1/(x+7) + 2/(x+3) = -4/((x+7)*(x+3))}}}
.
Now suppose we multiply the first term on the left side by {{{(x+3)/(x+3)}}} and the second
term on the left side by {{{(x+7)/(x+7)}}}.
.
Note that since the two multipliers each have the same numerator as they do denominator.
Therefore, they are each equivalent to 1.  So we are multiplying both terms by 1 and are
not changing the equation. The multiplication of the two terms makes the equation become:
.
{{{(1*(x+3))/((x+7)*(x+3)) +(2*(x+7))/((x+3)*(x+7)) = -4/((x+7)*(x+3))}}}
.
In this equation, every term has the denominator {{{(x+3)*(x+7)}}}. So we can get rid
of the denominators entirely by multiplying both sides of the equation (all terms) by
{{{(x+3)*(x+7)}}}. This in effect cancels the denominators out and the equation simplifies
to just the numerators … or to:
.
{{{1*(x+3) + 2*(x+7) = -4}}}
.
Do the multiplications on the left side and the equation becomes:
.
{{{x + 3 + 2x + 14 = -4}}}
.
Combine the terms on the left side to get:
.
{{{3x + 17 = -4}}}
.
Subtract 17 from both sides to reduce the equation to:
.
{{{3x = -21}}}
.
And solve for x by dividing both sides by 3 to get:
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{{{x = -21/3 = -7}}}
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All those who think x = -7 is the right answer, take one step forward. Not so fast ...
.
Go back to the original problem and look at {{{y[1] = 1/(x+7)}}}
.
What happens to the denominator of {{{y[1]}}} if you substitute -7 for x. The denominator
becomes zero … and division by zero is not allowed in algebra.  Therefore, our answer
of x = -7 will not work.  So the answer to this problem is that there is no value for
x that will satisfy this problem.
.
Hope this helps you get on to bigger and better problems than this one … and hope that
you work through all three of the answers you got and pick the one that seems to work for you.

An added note … according to the rules of algebra, when you write 1/x +7 you mean {{{1/x +7}}}
because the rules say to do multiplications and divisions first, then go back and do
the additions and subtractions. To make sure both the x and the +7 are in the denominator,
you need to write 1/(x+7). Then when you do the division you divide by the quantity (x+7).
The way you wrote it is a very common practice in this forum, but it often makes it difficult
to interpret the problem correctly. I hope I got your problem interpreted correctly in form …
.
```