http://www.purplemath.com/modules/solvelin3.htm
Most linear equations require
more than one step for their solution. For instance:
I need to undo the "times
seven" and the "plus two". There is no rule about which
"undo" I should do first. However, if I first divide through
by 7,
I'm going to have fractions. Personally, I prefer to avoid fractions
if possible, so I almost always do any plus / minus before any times
/ divide:
Then the solution is
x
= –8.
Formatting your homework
and showing your work in the manner I have done above is, in my experience,
fairly universally acceptable. However (warning!), it is also a good idea
to clearly rewrite your final answer at the end of each exercise, as shown
(in purple) above. Don't expect your grader to take the time to dig through
your work and try to figure out what you probably meant your answer to
be. Format
your work so as to make your meaning
clear!
- Solve –5x
– 7 = 108
Then the solution is
x
= –23.
- Solve 3x
–
9 = 33
Then the solution is
x
= 14.
- Solve 5x
+ 7x = 72
First, I need to combine
like terms on the left; then I can solve:
Then the solution is
x
= 6.
I need to move all the
x's
over to one side or the other. To avoid negative coefficients on my
variables, I usually move the smaller x;
in this case, I'll subtract the 4x
over to the other side:
Then the solution is
x
= –3.
In the above exercise,
note that it is perfectly okay to have the "x="
be on the right. The variable is not "required" to be on the
left; we're just used to seeing it there. It's alright if your solution
works out with the variable on the right. However (warning!), I have heard
of some instructors who insist that the variable be placed on the left-hand
side in the final
answer. (No, I'm
not making that up.) If you have any doubts about your instructor's formatting
preferences, ask now.
- Solve 8x
–
1 = 23 – 4x
Then the solution is
x
= 2.
- Solve 5
+ 4x –
7 = 4x – 2 – x
Before I can solve, I
need to combine like terms:
Then the solution is
x
= 0.
It is perfectly fine for
x
to have a value of zero. Zero is a valid solution. Do not say that this
equation has "no solution"; it does indeed have a solution,
that solution being x
= 0.
- Solve 0.2x
+ 0.9 = 0.3 – 0.1x
This equation solves just
like all the other linear equations. It just looks worse because
of the decimals. But that's easy to fix: however many decimal places
I have, I can multiply by "1"
followed by that number of zeroes. In this case, I'll multiply through
by 10:
10(0.2x)
+ 10(0.9) = 10(0.3) – 10(0.1x)
2x
+ 9 = 3 – 1x
Then I solve as usual:
2x
+ 1x + 9 – 9 = 3 – 9 – 1x + 1x
3x
= –6
x
= –2
If one of the decimals had
had two decimal places, then I'd have multiplied through by 100;
for three, I'd have multiplied through by 1000.
- Solve
To simplify my computations
for equations with fractions, I can first multiply through by the common
denominator. For this equation, the common denominator is 12:
3x
+ 12 = 2x
+ 6
3x
– 2x
+ 12 – 12 = 2x
– 2x
+ 6 – 12
x
= –6
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