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Beginning Algebra
Tutorial 14:
Solving Linear Equations

Learning Objectives

After completing this tutorial, you should be able to:
  1. Solve linear equations by using a combination of simplifying and usingvarious properties of equality.


In Tutorial 12:The Addition Propertyof Equality we looked at using the addition property ofequalityto help us solve linear equations.  In Tutorial13: The Multiplication Property of Equality we looked at usingthe multiplication property of equality and also put these two ideastogether. In this tutorial we will be solving linear equations by using acombinationof simplifying and various properties of equality. 

Knowing how to solve linear equations will open the doorto being ableto work a lot of other types of problems that you will encounter inyourvarious algebra classes.  It is very important to have thisconceptdown before moving ahead.  Make sure that you do not savor themystery of finding your variable, but work through some of these typesof problems until you have this concept down.


Strategy for Solving a LinearEquation

Note that your teacher or thebook you areusing may have worded these steps a little differently than I do, butitall boils down to the same concept -   get your variable ononeside and everything else on the other using inverse operations.

Step 1: Simplify each side, if needed.

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

To remove ( ):  Just use the distributiveproperty found in Tutorial 8: Properties of Real Numbers.

To remove fractions: Since fractions areanother way to writedivision, and the inverse of divide is to multiply, you removefractionsby multiplying both sides by the LCD of all of your fractions.  Ifyou need a

Step 2: Use Add./Sub. Properties tomove the variableterm to one side and all other terms to the other side. 

Step 3: Use Mult./Div. Properties toremove any valuesthat are in front of the variable.

Step 4:  Check your answer.

I find this is the quickest andeasiest wayto approach linear equations.

Example1:  Solve the equation .

*Inverse of add. 10 is sub. 10

*Inverse of mult. by -3 is -3

Be careful going from line 4to line 5. Yes, there is a negative sign. But, the operation between the -3 and xis multiplication not subtraction.  So if you were toadd3 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 youwill see that 1is the solution we are looking for.

Example2:  Solve the equation .

*Remove ( ) by using dist.prop.

*Get all xtermson one side

*Inverse of add. 3 is sub. 3

*Inverse of mult. by -1 is -1

If you put 9 back in for x in the original problem youwill see that9 is the solution we are looking for.

Example3:   Solve the equation..

*To get rid of thefractions, 
mult. both sides by the LCD of 4

*Get all the xterms on one side

*Inverse of add. 2 is sub. 2

*Inverse of mult. by -3 is -3


If you put 4/3 back in for x in the original problemyou will see that 4/3is the solution we are looking for.

Example4:   Solve the equation .

*To get rid of the decimals,
mult. both sides by 100

*Get all the yterms on one side

*Inverse of sub. 20 is add 20

*Inverse of mult. by 20 is 20


If you put 3/2 back in for y inthe originalproblem you will see that 3/2 is the solution we are looking for.


A contradiction is an equation with one variable thathas no solution.

Example5:   Solve the equation .


*Remove ( ) by using dist.prop.

*Get all the xterms on one side

Where did our variable, x,go??? It disappeared on us.  Also note how we ended up with a FALSEstatement, -1 is not equal to 12.  This does not mean that x= 12 or x = -1. 

Whenever your variable dropsout AND you endup with a false statement, then after all of your hard work, there isNOSOLUTION.

So, the answer is no solution.


An identity is an equation with one variable that hasall numbers asa solution.

Example6:   Solve the equation .


*Remove ( ) by using dist.prop.

*Get all the xterms on one side

This time when our variabledropped out, weended up with a TRUE statement.  Whenever that happens your answeris ALL REAL NUMBERS.

So, the answer is all real numbers.

Practice Problems

These are practice problems to help bring you to thenext level. It will allow you to check and see if you have an understanding ofthesetypes of problems. Math works just likeanythingelse, if you want to get good at it, then you need to practiceit. Even the best athletes and musicians had help along the way and lots ofpractice, 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 theproblem out onyour own and then check your answer by clicking on the link for theanswer/discussionfor that  problem.  At the link you will find the answeras well as any steps that went into finding that answer.



PracticeProblems 1a - 1d: 

Solve the given equation.

(answer/discussionto 1a)
(answer/discussionto 1b)

(answer/discussionto 1c)
(answer/discussionto 1d)

Need Extra Help on These Topics?

The following is a webpagethat can assistyou in the topics that were covered on this page: 

This webpage helps you with solving linear equations.


forsomemore suggestions.



All contents
June 22, 2003