**Learning Objectives**

After completing this tutorial, you should be able to:

- Use the multiplication and division properties of equalities to solve linear equations.
- Solve an equation using more than one property.
- Know how to express consecutive integers in terms of
*x*, if the first integer is*x*. - Know how to express even consecutive integers in terms of
*x*, if the first even integer is*x*. - Know how to express odd consecutive integers in terms of
*x*, if the first odd integer is*x*.

** Introduction**

As mentioned in **Tutorial
12: Addition
Property of Equality**, solving equations is getting into the
heart
of what algebra is about. As we did in Tutorial 12, we will
be looking specifically at linear equations and their solutions.
We will start off slow and solve equations that use only the
multiplication
or division property of equality 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.

** Tutorial**

The following definitions for **equation, linear
equation, solution
and solution set **can also be found in **Tutorial
12: The Addition Property of Equality**.

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)

If you need to review **what
a solution to an equation is, go to Tutorial 4: Introduction to
Variable
Expressions and Equations.**

Set of all solutions

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

In** Tutorial 12:
The Addition Property
of Equality**, we showed you how to solve equations using the **addition
and subtraction properties of equality**. In this tutorial we
will
be looking at the multiplication and division properties.

*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.

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.**

If you put -6 back in for *a *in
the original
problem you will see that **-6 is the solution we are looking for.**

Note that it doesn’t matter what side the variable is
on. -6 = *a* means the same thing as *a* = -6.

**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.**

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.**

What it boils down to is that
you want to get
the variable you are solving for alone on one side and everything else
on the other side using INVERSE operations.

** Example
4**: Solve the equation .

***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.**

***Inverse of sub. 5 is add 5**

***Inverse of mult. by -1 is div.
by -1**

If you put -2 back in for *x *in
the original
problem you will see that **-2 is the solution we are looking for.**

***Inverse of sub. 2 is add 2**

***Inverse of mult. by 7 is div.
by 7**

If you put 2 back in for *x *in
the original
problem you will see that **2 is the solution we are looking for.**

For example, 5, 6, and 7 are three
consecutive integers.

If we let *x* represent the first integer,
how would we represent the second consecutive integer in terms of* x*? Well if we look at 5, 6, and 7 - note that 6 is one
more than 5, the first integer.

In general, **we could represent the second
consecutive integer by x + 1**. And what about the third consecutive integer.

Well, note how 7 is 2 more than 5. In
general, **we could represent
the third consecutive integer as x + 2.**

**Consecutive EVEN integers **are even integers that
follow one another
in order.

For example, 4, 6, and 8 are three consecutive
even integers.

If we let *x* represent the first EVEN integer,
how would we represent the second consecutive even integer in terms of *x*?
Note that 6 is two more than 4, the first even integer.

In general, **we could represent the second
consecutive EVEN integer
by x + 2**.

And what about the third consecutive even
integer? Well, note
how 8 is 4 more than 4. In general, **we could represent the
third
consecutive EVEN integer as x + 4.**

**Consecutive ODD integers **are odd integers that
follow one another
in order.

For example, 5, 7, and 9 are three consecutive
odd integers.

If we let *x* represent the first ODD integer,
how would we represent the second consecutive odd integer in terms of *x*?
Note that 7 is two more than 5, the first odd integer.

In general, **we could represent the second
consecutive ODD integer
by x + 2.**

And what about the third consecutive odd
integer? Well, note how
9 is 4 more than 5. In general, **we could represent the third
consecutive
ODD integer as x + 4.**

Note that a common misconception is that because
we want an odd number
that we should not be adding a 2 which is an even number. Keep in
mind that *x* is representing an ODD
number and
that the next odd number is 2 away, just like 7 is 2 away form 5, so we
need to add 2 to the first odd number to get to the second consecutive
odd number.

If *x* represents the first
of four consecutive
integers, express the sum of the four integers in terms of *x.*

First of all, we need to have **all four consecutive
integers in terms of ***x*.

We can represent them the following way:

*x* + 1
= 2nd consecutive integer

*x *+
2 = 3rd consecutive integer

*x* + 3
= 4th consecutive integer

Second we need to **write it as a sum of the four
integers and then
simplify it**:

If *x* represents the first
of three odd consecutive
integers, express the sum of the first and third integers in terms of *x*.

First of all, we need to have **all three consecutive
ODD integers in terms of ***x*.

We can represent them the following way:

*x* + 2
= 2nd consecutive odd
integer

*x *+
4 = 3rd consecutive odd
integer

Second we need to write it as **a sum of the first and
third odd integers
in terms of x and then simplify it:**

** 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.

Practice Problem 2a:Write an algebraic expression and simplify if possible.

2a. If *x* represents the first of
three consecutive integers, express the sum of the three integers in
terms
of *x*.

(answer/discussion to 2a)

(answer/discussion to 2a)

** Need Extra Help on these Topics?**

**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 26, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.