Intermediate Algebra
Tutorial 7:
Linear Equations in One
Variable
 Learning Objectives
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After completing this tutorial, you should be able to:
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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.
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Know when an equation has no solution.
-
Know when an equation has all real numbers as a solution.
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Introduction
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| 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.
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Tutorial
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|
Equation
Two expressions set equal to each other
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Linear Equation
An equation that can be written in the form
ax + b = c
where a, b, and c are constants
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| The following is an example of a linear equation:
3x - 4 = 5
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Solution
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)
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Solution Set
Set of all solutions
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 Example
1: Determine if any of the following values for x are
solutions
to the given equation. 3x - 4
=
5 ; x = 3, 5. |
Checking 3
3x - 4 = 5
3(3) - 4 = 5
9 - 4 = 5
5 = 5
True
3
is a solution |
Checking 5
3x - 4 = 5
3(5) - 4 = 5
15 - 4 = 5
11 = 5
False
5
is not a solution |
Solving a Linear Equation
in General
Get the variable you are solving for alone on one side
and everything
else on the other side using INVERSE operations.
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| The following will give us the tools that we need to
solve linear equations. |
|
Addition and Subtraction
Properties of Equality
If a = b, then a + c = b + c
If a = b, then a - c = b - c
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| In other words, if two expressions are equal to each
other and you
add or subtract the exact same thing to both sides, the two sides will
remain equal.
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.
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x - 5 = 2
x - 5 + 5 = 2 + 5
x = 7 |
*Inverse of sub. 5 is add.
5 |
| 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. |
| Example
3: Solve for the variable. y
+ 4 = -7. |
y + 4 = -7
y + 4 - 4 = -7 - 4
y = -11 |
*Inverse of add. 4 is sub. 4 |
| 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. |
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Multiplication and Division
Properties of Equality
If a = b, then a(c) = b(c)
If a = b, then a/c = b/c where c is
not equal to 0.
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| 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.
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 |
*Inverse of div. by 2 is
mult. by 2 |
| If you put 10 back in for x in
the original
problem, you will see that 10 is the solution we are looking for. |
| Example
5: Solve for the variable. 5x
= 7. |
 |
*Inverse of mult. by 5 is div.
by 5 |
| 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. |
| The examples above were using
only one property
at a time to help you understand the different properties that we use
to
solve equations. However, most times, we have to use several
properties
to get the job done. The following is a strategy that you can use
to help you solve linear equations that are a little bit more involved. |
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Strategy for Solving a Linear
Equation
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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.
| 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.
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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
- 3x = 7.
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 |
*Inverse of add. 10 is sub. 10
*Inverse of mult. by -3 is div.
by -3
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| 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.
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| Example
7: Solve for the variable. 2(x
+ 5) - 7 = 3(x - 2). |
 |
*Remove ( ) by using dist.
prop.
*Get all x
terms
on one side
*Inverse of add. 3 is sub. 3
*Inverse of mult. by -1 is div.
by -1
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| 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
8: Solve for the variable:  . |
 |
*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
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| 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. |
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Contradiction
A contradiction is an equation with one variable that
has no solution.
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| Example
9: Solve for the variable. 4x
- 1 = 4(x + 3). |
 |
*Remove ( ) by using dist. prop.
*Get all the x
terms on one side
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| 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.
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Identity
An identity is an equation with one variable that has
all numbers as
a solution.
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| Example
10: Solve for the variable. 5x
+ 10 = 5(x + 2). |
 |
*Remove ( ) by using dist. prop.
*Get all the x
terms on one side
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| 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.
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Practice Problems
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| 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.
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 Practice
Problems 1a - 1e:
Solve for the variable.
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Need Extra Help on These Topics?
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 22, 2003 by Kim Seward. |
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