Learning Objectives
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
Step 1: Simplify each side, if needed.
To remove ( ): Just use the distributive property found in Tutorial 8: 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. If you need a review on the LCD, go to Tutorial 3: 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 1: 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.
*Get all x terms on one side
*Inverse of add. 3 is sub. 3
*Inverse of mult. by -1 is div.
by -1
*Get all the x terms on one side
*Inverse of add. 2 is sub. 2
*Inverse of mult. by -3 is div. by -3
*Get all the y terms on one side
*Inverse of sub. 20 is add 20
*Inverse of mult. by 20 is div. by 20
A contradiction is an equation with one variable that has no solution.
*Get all the x terms on one side
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 numbers as
a solution.
*Get all the x terms on one side
So, the answer is all real numbers.
Practice Problems
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?
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.