**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 and subtraction properties of equalities to solve linear equations.

** 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. In this and the next
tutorial,
we will start off slow and solve equations that use only one property
to
make sure you have the individual concepts down. Then, in later
tutorials,
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. The tutorial is ready when you are.

** 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)

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.

*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. **

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.**

***LCD = 4**

***1/2 = 2/4**

If you put -1/4 back in for *y *in
the original
problem you will see that **-1/4 is the solution to our problem.**

In this problem, our variable *a *is
on both
sides of the equation. As mentioned above, **when solving a
linear
equation you need to get the variable you are
solving
for alone on one side and everything else on the other side
using
INVERSE operations.**

At this point we are limited. We only have talked about using the addition and subtraction properties of equality. In Tutorial 13, we will address the multiplication and division properties of equality. But since this was made before that, we have to make ado with addition and subtraction.

We can solve this with what we know so far. We
move a term that
has a variable exactly the same way we were moving constants in
examples
1 and 2. In this problem we need to get *a* on one side and everything else on the other. We have a -.7*a* on the right side. To move it to the other side, so *a* is only on one side, we will do the inverse of minus, which is add .7a
to both sides.

After that it looks like examples 1 and 2 above, and we
continue doing
inverse operations until we have *a* on
one side
and everything on the other side of the equation.

**Let's see what we get:**

***Inverse of add 1.2 is
sub. 1.2 **

If you put -3.6 back in for *a* in the original
problem you will see that **-3.6 is the solution to our problem.**

In this problem, we need to simplify the left side
first before we
can start adding or subtracting things around. We can accomplish
this by using the **distributive
property (found in Tutorial 8: Properties of Real Numbers)**.

**Using the distributive property and then combining
like terms
to simplify the left side of the equation we get:**

***Inverse of sub. 14 is add.
14 **

If you put 15 back in for *x *in
the original
problem you will see that **15 is the solution to our problem.**

Let’s put this one in terms that everyone can relate
to, MONEY.
Let's say that you owe two people a total of $100. You owe the
first
person $75. How much do you owe the second person? The
answer
would be $100 - $75 = $25. To figure it out you would take the
total
and then subtract out the known amount to get the other amount.

We can use that concept to figure out our problem. Anytime you know the total of two numbers, you subtract the given from the total to either find the other number or express the other number in terms of a variable.

Since our total is 100 and we are letting *x* represent one number, the other number would be expressed as the total
minus *x* or 100 -* x*.

So, **100 - x is our
answer.**

** 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.

2a. Two numbers have a sum of 200. If one
number is *x*,
express the other number 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.

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