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Beginning Algebra Tutorial II--Evaluating Expressions and Solving Equations

Updated on September 2, 2012
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Welcome

My first tutorial, Beginning Algebra Tutorial I--Exponents and PEMDAS covered the very basics of Exponents and the Order of Operations (PEMDAS).

In this tutorial, I will explain how to evaluate very basic expressions and solve very basic equations.

Algebra can be difficult for many people and I hope my tutorials shed a light to those that are learning for the first time or serve to help those that have not used Algebra in a long time.


Expression Vs. Equations--What's the difference?

Equations

An Equation does

contain the = sign.

A few examples of equations:

  • 5x + 2 = 32
  • x/28 = 7
  • 8x = 64

Expressions

An Expression does not

contain the = sign.

A few examples of expressions:

  • x + y
  • 2x
  • 5/y

Evaluating Expressions

Let's begin with evaluating expressions. Expressions involve mathematical operations on a combination of numbers and variables. Common variables are x, y, and z, as well as a, b, and c, but many different letters are used to represent a variable depending on the problem.

When asked to evaluate an expression you simply replace x or other variables with the new value/values given.

Evaluate: 2x

If x = 5

  • First, understand the problem.
  • 2x means 2(x) or 2 • x
  • Next look at the information we are given if x = 5
  • Now wherever we see an x we simply replace x with the number 5.
  • We now have 2(5) or 2 • 5
  • Now solve 2 • 5 = 10

In this problem we know if x = 5 the expression evaluates to 10.

Evaluate: 3y + x

If y = 2 and x = 10

  • First, understand the problem.
  • 3y means 3(y) or 3 • y
  • Wherever we see a y we replace y with the number 2.
  • We now have 3(2) + x.
  • Wherever we see a x we replace x with the number 10.
  • We now have 3(2) + 10.
  • 3 • 2 = 6
  • Now solve 6 + 10 = 16.

In this problem we know if y = 2 and x = 10 the expression evaluates to 16.

Evaluate: 30/y + 2x + 10

If y = 5 and x = 4

  • First, understand the problem.
  • Wherever we see a y we replace y with the number 5.
  • We now have 30/5 + 2x + 10.
  • 30/5 = 6
  • We now have 6 + 2x + 10.
  • Wherever we see a x we replace x with the number 4.
  • 2(4) = 2 * 4 = 8.
  • We now have 6 + 8 + 10
  • Now solve 6 + 8 + 10 = 24.

Evaluate: 96/x - 12y - 3

If x = 12 and y = 4

  • First, understand the problem.
  • Wherever we see a x we replace x with the number 12.
  • We now have 96/12 - 12y - 3.
  • 96/12 = 8.
  • We now have 8 - 12y - 3.
  • Wherever we see a y we replace y with the number 4.
  • 12(4) = 12 * 4 = 48.
  • We now have 8 - 48 - 3
  • Now solve 8 - 48 - 3 = - 43

Solving Equations

Many people can solve simple math problems involving numbers, however once variables are added into the mix, some are baffled.

Properties of Equality

These properties allow us to manipulate equations to help us solve.

The same number can be added, subtracted, multiplied, or divided on each side.

What is done to one side of the equation has to be done on the other side of the equation.

Addition

  • a = b then a + c = b + c Addition property c is added to both sides.

Subtraction

  • a = b then a - c = b - c Subtraction property c is subtracted from both sides.

Multiplication

  • a = b then a * c = b * c Multiplication property c is multiplied on both sides.

Division

  • a = b then a/c = b/c Division property c is divided on both sides.

Simple Example

I will begin with a simple math problem.

We should know that 2 + 3 = 5.

Since we already know the above problem, let's throw in a variable to the same equation and solve that same problem.

2 + x = 5


Due to the Properties of Equality we are able to manipulate the equation to solve for x.

To solve the equation we want to have the variable x by itself.

In order to have x by itself, we need to subtract the number 2 from both sides because what we do to one side of the equation we need to do to the other side of the equation.

  • 2 + x = 5
  • Subtract the number 2 from both sides. 2 - 2 + x = 5 - 2
  • 2 - 2 = 0
  • We now have x = 5 - 2
  • 5 - 2 = 3
  • x = 3


Notice that 3 is the same number we replaced with the variable x. This was just a simple example to give you an idea on how to solve a very basic equation.

Problem One: 5x - 2 + 4x = 16

  • First combine like terms. 5x + 4x = 9x
  • Now we have 9x - 2 = 16
  • We want to have the x by itself. First add 2 to both sides.
  • 9x - 2 + 2 = 16 + 2
  • 9x + 0 = 18
  • In order to have the x by itself we will need to undo the multiplication of 9x. To undo the multiplication we will need to divide both sides by 9.
  • 9x/9 = 18/9 On the left side the 9s cancel out.
  • x = 2
  • Simplify the right side 18/9
  • x = 2

Check Work

In the above equation x = 2.

5x - 2 + 4x = 16

To check work, wherever there is an x, replace the x with the number 2.

  • 5(2) - 2 + 4(2) = 16
  • 5 * 2 = 10 and 4 * 2 = 8
  • 10 - 2 + 8 = 16
  • Simplify 10 - 2 + 8 = 16
  • 8 + 8 = 16
  • 16 = 16
  • True 16 does equal 16. So the answer is Correct!

Problem Two: x/28 = 7

This problem is actually really simple. We need to have x by itself. All we have to do is undo the division. In order to undo the division we need to multiply 28 by both sides.

Check Work

In the above equation x = 196.

x/28 = 7

  • Wherever there is an x, replace the x with 196.
  • 196/28 = 7

This equation is Correct.

Problem Three: 2x - 4 = -3(x + 12)

This may seem tricky at first. Going by the Order of Operations, Parenthesis come first.

  • We will use the distributive property to remove the Parenthesis.

  • We now have 2x - 4 = -3x - 36
  • We need to have the x variables on one side. So we will add 3x to both sides.
  • 2x + 3x - 4 = -3x + 3x - 36
  • Combine like terms 2x + 3x = 5x and -3x +3x = 0
  • We now have 5x - 4 = - 36
  • We need to have x by itself so we need to add 4 to both sides.
  • 5x - 4 + 4 = -36 + 4
  • 5x = -32
  • Divide both sides by 5. 5x/5 = -32/5
  • On the left side the 5s cancel out.
  • x = -32/5

Check Work

In the above equation x = - 32/5.

2x - 4 = -3(x + 12)

2x - 4 = -3x - 36 (After using the Distributive Property)


To check work, wherever there is an x replace the x with -32/5.

  • 2(-32/5) - 4 = -3(-32/5) - 36
  • 2(-32/5) - 4 = -84/5
  • -3(-32/5) - 36 = -84/5
  • -84/5 = -84/5
  • True -84/5 = -84/5 so the answer is Correct.

Final Thoughts

I hope that this tutorial gave you a very basic understanding of the basic rules of Evaluating Expressions as well as Solving basic Equations.

For those just learning, remember Math can be hard when first learned, but with practice and determination...Anything is possible!!!

Thank-you for taking the time to read this article. GOOD LUCK and NEVER GIVE UP!!!

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    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      IntegrityYes, You are such a supportive and kind person to know!!! I definitely love that regardless of genre, you always take the time to read and comment!!! Thank-you!!!!

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      Mama Kim, I absolutely LOVE math and want to help others understand and love it too!!! :)

      Thank-you for taking the time to read and comment!!! I always do appreciate it very much!!!!

    • profile image

      IntegrityYes 5 years ago

      You are very welcome, Patty

    • Mama Kim 8 profile image

      Sasha Kim 5 years ago

      I took algebra in middle school and LOVED it! One of my all time favorite classes. ^_^ I took each problem to be a puzzle and just had a blast solving them. The way you explain it in your hub is perfect! wonderful, voted up and shared.

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      IntegrityYes, Thank-you soooo much for your continued support and encouragement!!!! I truly do appreciate it!!!!

    • profile image

      IntegrityYes 5 years ago

      Many need to read this often. Rock on, Patty!

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      tjdavis, Thank-you sooooo very much for your comments!!! Because school has started in most areas or will be starting soon, I will have more tutorials soon. I wish your daughter the very best with Algebra!!! THANKS AGAIN!!!!

    • tjdavis profile image

      Teresa Davis 5 years ago from Moscow, Texas

      Loved this hub..very informative. I'm going to share this with my daughter who is doing Algebra. You wrote it so even I could keep up lol..Thanks!!!

    • Patty Kenyon profile image
      Author

      Patty Kenyon 5 years ago from Ledyard, Connecticut

      Deergha, Thank-you sooooo much for your encouraging comments!!! Math is hard for soooo many and I try to think of easy ways of explaining it!!! Thank-you for the Votes!!! I do appreciate it!!!

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      Michelle, I was thinking the same thing in regards to the coincidence...lol...I loved your Hub on teaching Literature!!! Awesome Ideas!!! Thank-you soooo much for the votes and share!!! I do appreciate it!!!

    • dghbrh profile image

      deergha 5 years ago from ...... a place beyond now and beyond here !!!

      Dear Patty,

      I used to love algebra a lot during school. Now I am not into algebra that much except when I teach something to son. Your tutorial is very simple ans easy to understand. You have done really a very nice job here.

      Thanks for sharing this one.

      Votes up and away.

      Deergha

    • midget38 profile image

      Michelle Liew 5 years ago from Singapore

      Hey Patty, thanks for the Maths lesson. What a coincidence that we both wrote about something to do with school and learning today! Now, you should've been my maths tutor. But I love algebra!! Votes up and away and shared.

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      TToombs, Thank-you soooo very much!!! I recently was back in school and Algebra was the hardest for sooo many of my fellow classmates both those fresh out of high school as well as adult students. I figured what knowledge I gained, I would pass along!!!

      Thank-you for taking the time to read and comment!!! I do appreciate your support!!!

    • TToombs08 profile image

      Terrye Toombs 5 years ago from Somewhere between Heaven and Hell without a road map.

      Patty, great tutorial on algebra! I could have used this instead of my 2 trips through algebra in college. It could have saved me hundreds of dollars! :) Great job. Voted up and more. :)

    • Patty Kenyon profile image
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      Patty Kenyon 5 years ago from Ledyard, Connecticut

      Frank, lol...Math is really hard for so many. Last school year, because of budget cuts, my daughters' high school had a history teacher teaching math so I helped tutor some of their friends because the kids didn't understand what the teacher was doing.

      Thank-you for taking the time to read this and comment!!! I truly do appreciate your support!!!!! :)

    • Frank Atanacio profile image

      Frank Atanacio 5 years ago from Shelton

      Okay Patty so you're smart.. so what? where the hell were you when I was a freshman in high school.. man math really kicked my butt.. but this will help others and thank you for sharing :)