Example 1 Solve
each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
In the following problems we will describe in detail the first
problem and the leave most of the explanation out of the following problems.
(a)
For this problem there are no fractions so we don’t need
to worry about the first step in the process.
The next step tells to simplify both sides. So, we will clear out any parenthesis by
multiplying the numbers through and then combine like terms.
The next step is to get all the x’s on one side and all the numbers on the other side. Which side the x’s go on is up to you and will probably vary with the
problem. As a rule of thumb we will
usually put the variables on the side that will give a positive
coefficient. This is done simply
because it is often easy to lose track of the minus sign on the coefficient
and so if we make sure it is positive we won’t need to worry about it.
So, for our case this will mean adding 4x to both sides and subtracting 15
from both sides. Note as well that
while we will actually put those operations in this time we normally do these
operations in our head.
The next step says to get a coefficient of 1 in front of
the x. In this case we can do this by dividing
both sides by a 7.
Now, if we’ve done all of our work correct is the solution to the equation.
The last and final step is to then check the
solution. As pointed out in the
process outline we need to check the solution in the original equation. This is
important, because we may have made a mistake in the very first step and if
we did and then checked the answer in the results from that step it may seem to
indicate that the solution is correct when the reality will be that we don’t
have the correct answer because of the mistake that we originally made.
The problem of course is that, with this solution, that
checking might be a little messy.
Let’s do it anyway.
So, we did our work correctly and the solution to the
equation is,
Note that we didn’t use the solution set notation
here. For single solutions we will
rarely do that in this class. However,
if we had wanted to the solution set notation for this problem would be,
Before proceeding to the next problem let’s first make a
quick comment about the “messiness’ of this answer. Do NOT expect all answers to be nice simple
integers. While we do try to keep most
answer simple often they won’t be so do NOT get so locked into the idea that
an answer must be a simple integer that you immediately assume that you’ve
made a mistake because of the “messiness” of the answer.
[Return to Problems]
(b)
Okay, with this one we won’t be putting quite as much
explanation into the problem.
In this case we have fractions so to make our life easier
we will multiply both sides by the LCD, which is 21 in this case. After doing that the problem will be very
similar to the previous problem. Note
as well that the denominators are only numbers and so we won’t need to worry
about division by zero issues.
Let’s first multiply both sides by the LCD.
Be careful to correctly distribute the 21 through the
parenthesis on the left side.
Everything inside the parenthesis needs to be multiplied by the 21
before we simplify. At this point
we’ve got a problem that is similar the previous problem and we won’t bother
with all the explanation this time.
So, it looks like is the solution. Let’s verify it to make sure.
So, it is the solution.
[Return to Problems]
(c)
This one is similar to the previous one except now we’ve
got variables in the denominator. So,
to get the LCD we’ll first need to completely factor the denominators of each
rational expression.
So, it looks like the LCD is . Also note that we will need to avoid since if we plugged that into the equation
we would get division by zero.
Now, outside of the y’s
in the denominator this problem works identical to the previous one so let’s
do the work.
Now the solution is not so we won’t get division by zero with the
solution which is a good thing.
Finally, let’s do a quick verification.
So we did the work correctly.
[Return to Problems]
(d)
In this case it looks like the LCD is and it also looks like we will need to avoid
and to make sure that we don’t get division by
zero.
Let’s get started on the work for this problem.
At this point let’s pause and acknowledge that we’ve got a
z^{2} in the work
here. Do not get excited about
that. Sometimes these will show up
temporarily in these problems. You
should only worry about it if it is still there after we finish the
simplification work.
So, let’s finish the problem.
Notice that the z^{2}
did in fact cancel out. Now, if we did
our work correctly should be the solution since it is not
either of the two values that will give division by zero. Let’s verify this.
The checking can be a little messy at times, but it does
mean that we KNOW the solution is correct.
[Return to Problems]
