**Learning Objectives**

After completing this tutorial, you should be able to:

- Know what a linear equation is.
- Know if a value is a solution or not.
- Use the addition, subtraction, multiplication, and division properties of equalities to solve linear equations.
- Know when an equation has no solution.
- Know when an equation has all real numbers as a solution.

** Introduction**

This is where we start getting into the heart of what algebra is about - solving equations. In this tutorial we will be looking specifically at linear equations and their solutions. We will start off slow and solve equations that use only one property to make sure you have the individual concepts down. Then we will pick up the pace and mix 'em up where you need to use several properties and steps to get the job done.

Equations can be used to help us solve a variety of problems. In later tutorials, we will put them to use to solve word problems. Then you can answer those tricky math questions.

** Tutorial**

Two expressions set equal to each other

An equation that can be written in the form

*ax + b = c*

*where a, b, and c are constants*

The following is an example of a linear equation:
3*x* - 4 = 5

A value, such that, when you replace the variable with
it,

it makes
the equation true.

(the left side comes out equal to the right side)

Set of all solutions

3

3

3(3) - 4 = 5

9 - 4 = 5

5 = 5

3

3(5) - 4 = 5

15 - 4 = 5

11 = 5

Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations.

The following will give us the tools that we need to
solve linear equations.

*If a = b, then a + c = b + c*

*If a = b, then a - c = b - c*

Note that addition and subtraction are inverse
operations of each
other. For example, if you have a number that is being added that
you need to move to the other side of the equation, then you would
subtract
it from both sides of that equation.

** Example
2**: Solve for the variable. *x* - 5 = 2.

Note that if you put 7 back in for *x *in
the original problem you will see that **7 is the solution to our
problem.**

Note that if you put -11 back in for *y* in the original problem you will see that **-11 is the solution we
are
looking for**.

*If a = b, then a(c) = b(c)*

*If a = b, then a/c = b/c where c is
not equal to 0.*

In other words, **if two expressions are equal to
each other and you
multiply or divide (except for 0) the exact same constant to both
sides,
the two sides will remain equal. **

**Note that multiplication and division are inverse
operations of each
other. For example, if you have a number that is being multiplied
that you need to move to the other side of the equation, then you would
divide it from both sides of that equation. **

Note, for multiplication and division, it is not
guaranteed that if
you multiply by the variable you are solving for that the two sides are
going to be equal. But is guaranteed that the two sides are going
to be equal if you are multiplying or dividing by a constant or another
variable that you are not solving for. We will talk more about
this
in a later tutorial. For this tutorial just note you can use this
property with constants and variables you are not solving for.

** Example
4**: Solve for the variable. *x*/2
= 5.

If you put 10 back in for *x *in
the original
problem, you will see that **10 is the solution we are looking for.**

If you put 7/5 back in for* x* in the original
problem, you will see that **7/5 is the solution we are looking for.**

**Step 1: Simplify each side, if needed.**

This would involve things like removing ( ),
removing fractions, adding
like terms, etc.

**To remove ( ): ** Just use the **distributive
property found in Tutorial 5: Properties of Real Numbers.**

**To remove fractions**: Since fractions are
another way to write
division, and the inverse of divide is to multiply, you remove
fractions
by multiplying both sides by the LCD of all of your fractions.

**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
6**: Solve for the variable. 10 - 3*x* = 7.

***Inverse of mult. by -3 is div.
by -3**

If you put 1 back in for x in the original problem you
will see that **1
is the solution we are looking for.**

***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.**

***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.**

A contradiction is an equation with one variable that has no solution.

***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.**

An identity is an equation with one variable

that has
all real numbers as
a solution.

***Get all the x terms on one side**

So, **the answer is all real numbers**.

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 - 1e: Solve for the variable.

** Need Extra Help on these Topics?**

**http://www.purplemath.com/modules/solvelin.htm**

This webpage helps you to solve linear equations.

**http://www.math.com/school/subject2/lessons/S2U3L1DP.html**

This website helps you solve linear equations using either the addition
or subtraction properties of equality.

**http://www.math.com/school/subject2/lessons/S2U3L2DP.html**

This website helps you solve linear equations using the division
property
of equality.

**http://www.math.com/school/subject2/lessons/S2U3L3DP.html**

This website helps you solve linear equations using the multiplication
property of equality.

**http://www.mathpower.com/tut50.htm**

This webpage gives an example of solving a linear equation.

**http://www.sosmath.com/algebra/solve/solve0/solve0.html#linear**

This website helps you to solve linear equations. **Only
do the linear equations at this site. Do not go on to equations
containing
radicals - that will be covered in a later tutorial.**

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on July 1, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.