Beginning Algebra
Tutorial 14: Solving Linear Equations
Learning Objectives
After completing this tutorial, you should be able to:
 Solve linear equations by using a combination of simplifying and using
various properties of equality.

Introduction
In Tutorial 12:
The Addition Property
of Equality we looked at using the addition property of
equality
to help us solve linear equations. In Tutorial
13: The Multiplication Property of Equality we looked at using
the multiplication property of equality and also put these two ideas
together.
In this tutorial we will be solving linear equations by using a
combination
of simplifying and various properties of equality.
Knowing how to solve linear equations will open the door
to being able
to work a lot of other types of problems that you will encounter in
your
various algebra classes. It is very important to have this
concept
down before moving ahead. Make sure that you do not savor the
mystery of finding your variable, but work through some of these types
of problems until you have this concept down.

Tutorial
Strategy for Solving a Linear
Equation

Note that your teacher or the
book you are
using may have worded these steps a little differently than I do, but
it
all boils down to the same concept  get your variable on
one
side and everything else on the other using inverse operations.
Step 1: Simplify each side, if needed.
Step 2: Use Add./Sub. Properties to
move the variable
term to one side and all other terms to the other side.
Step 3: Use Mult./Div. Properties to
remove any values
that are in front of the variable.
Step 4: Check your answer.
I find this is the quickest and
easiest way
to approach linear equations.
Example
1: Solve the equation . 

*Inverse of add. 10 is sub. 10
*Inverse of mult. by 3 is div.
by 3 
Be careful going from line 4
to line 5.
Yes, there is a negative sign. But, the operation between the 3 and x is multiplication not subtraction. So if you were to
add
3 to both sides you would have ended up with 3x + 3 instead of the desired x.
If you put 1 back in for x in the original problem you
will see that 1
is the solution we are looking for. 
Example
2: Solve the equation . 

*Get all x terms
on one side
*Inverse of add. 3 is sub. 3
*Inverse of mult. by 1 is div.
by 1

If you put 9 back in for x in the original problem you
will see that 9 is the solution we are looking for. 
Example
3: Solve the equation .. 

*To get rid of the
fractions,
mult. both sides by the LCD of 4
*Get all the x terms on one side
*Inverse of add. 2 is sub. 2
*Inverse of mult. by 3 is div.
by 3

If you put 4/3 back in for x in the original problem
you will see that 4/3
is the solution we are looking for. 
Example
4: Solve the equation . 

*To get rid of the decimals,
mult. both sides by 100
*Get all the y terms on one side
*Inverse of sub. 20 is add 20
*Inverse of mult. by 20 is div.
by 20

If you put 3/2 back in for y in
the original
problem you will see that 3/2 is the solution we are looking for. 
Contradiction
A contradiction is an equation with one variable that
has no solution.

Example
5: Solve the equation . 

*Get all the x terms on one side

Where did our variable, x, go??? It disappeared on us. Also note how we ended up with a FALSE statement, 1 is not equal to 12. This does not mean that x = 12 or x = 1.
Whenever your variable drops
out AND you end
up with a false statement, then after all of your hard work, there is
NO
SOLUTION.
So, the answer is no solution. 
Identity
An identity is an equation with one variable
that has
all numbers as
a solution.

Example
6: Solve the equation . 

*Get all the x terms on one side

This time when our variable
dropped out, we
ended up with a TRUE statement. Whenever that happens your answer
is ALL REAL NUMBERS.
So, the answer is all real numbers. 
Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1d: Solve the given equation.
Need Extra Help on these Topics?
Last revised on July 26, 2011 by Kim Seward.
All contents copyright (C) 2001  2010, WTAMU and Kim Seward. All rights reserved.

