How to Learn Algebra Fast—Rules, Equations, Solutions

Updated on January 8, 2018
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This author has also written numbering system lessons for base 2, 4, 8, and 16. See separate "Person of Interest" profile page.

Here is your complete, free, beginner algebra and equations tutorial. It is recommended one does not attempt to do all at once; bookmark and return as desired.

If you already know arithmetic (including fractions and decimals), then you already know algebra. You just don't know you know yet. If you understand the answers to the following statements, proceed with this page; otherwise, probably not a good idea.

  • 5 + 7 = 12
  • 14 - 12 = 2
  • 4 times 8 = 32
  • 50 divided by 10 = 5
  • 50 divided by 40 = 1.25

The Basics

Example #1

Algebra is nothing more than merely substituting letters for numbers. As an example:

3 + 1 = 4

So, if we say the letter A is temporarily equal to 3, i.e.:

A = 3

And

The letter B is temporarily equal to 1, i.e.:

B = 1

Then A plus B must equal 4, i.e.:

A + B = 4

Example #2

A = 5

B = 2

So,

A + B = ?

Well, if we replace the letter A with 5, then the question becomes:

5 + B = ?

And then when we replace the letter B with 2, we have:

5 + 2 = ?

Problem solved.

A side note: Algebra likes to use the letter X in place of the question mark. So the correct way to have stated the above question would have been to say:

A = 5

B = 2

X = A + B

What is X?

The answer is:

X = 7

Congratulations, you have just learned the basic concept of algebra.

Example #3: Subtraction

A = 9

B = 4

X = A - B

What is X?

We plug in the numbers and we get:

X = 9 - 4

X = 5

Multiplication and Division

Of course multiplication and division in algebra are just the same as in arithmetic.

Multiplication Example

(The asterisk sign (“*”) is used to replace the word “multiply.”)

A = 20

B = 5

X = A * B

What is X?

We plug in the numbers and we get:

X = 20 * 5

X = 100

Division Example

(The “/” sign is used to replace the word “divide.”)

A = 20

B = 5

X = A/B

What is X?

We plug in the numbers and we get:

X = 20/5

X = 4

Let's Mix Things Up

You now know all the arithmetic functions of algebra. Algebra lets you mix up and combine these functions.

For example:

A=1

B=2

C=3

D=4

X=A+B+C+D

X=10

Let’s include subtraction:

X=A + B + C - D

X=(1+2+3) - 4, or

X = 6 - 4, which is 2, or

X = 6 - 4 = 2

Yes, there can be more than one equal sign in an equation. Instead of saying,

A=7

B=7

C=7

D=7

You can say,

A=7

A=B=C=D

Or just say,

A=B=C=D=7

Didn’t know you were doing equations, did you? You have been solving equations since the first paragraph.

Just Some Random Example NASA Formulas

In Algebra How Do You Solve for V? Basic / beginner algebra volume formulas.
In Algebra How Do You Solve for V? Basic / beginner algebra volume formulas. | Source

Note the "d²" in the volume formula for the cylinder. Yes, the upper "2" means the variable "d" is squared or itself times itself or "d" to the second power.

Note the "a³" in the volume formula for the cube. Likewise, the upper "3" means the variable "a" is cubed or itself times itself times itself or "a" to the third power.

Notice how some of the variables in the above formulas are directly adjacent to each other. This is the standard used to indicate the variables are multiplied.

Examples

  • The rectangular prism formula or equation, V = a b h, means volume is equal to "a" times "b" times "h".
  • The top half of the volume for the sphere formula or equation, "πd³", means pi times d after d has been cubed. If d was equal to 5, then d³ would equal 125, making the equation π times 125 or 125π.

Yes, the slash in the sphere and cylinder formulas means divide by that lower number, 6 and 4 respectively.

The approximate value of π is 3.14159, this value serves most everyday purposes just fine.

More Multiplication Practice

Example #1

Time to include multiplication.

A=1

B=2

C=3

D=4

X=A+B*C-D

What is X?

Simplify and solve.

When you see an equation has multiplication and division mixed into it, the rule is to do the multiplication and division first, then do the +’s and -‘s.

So the equation above really means,

X = A + (B*C) - D or

X = 1 + (2*3) - 4 or

X = 1 + (6) - 4

X = 3

The “(“ and the “)” are used to tell you what parts of the equation to do first.

It should be noted X=A and A=X are mathematically equal.

Just Like the Pros

What you have been and are doing is just simplifying, a.k.a breaking down, the equation one piece at a time; just like the mathematicians do it. The mathematicians are no more able to look at an equation and instantly come up with the answer any better than the rest of us can. In other words, they can’t grasp the whole equation either. They just solve and proceed from line to line, trusting they solved the previous line(s) correctly.

Example #2

Here is another one:

A=1, B=2, C=3, D=4, E=5, F=6

((D*B) + (F - 7)) + A) * C = X.

What is X?

This time there is more than one set of parentheses. When that happens, the rule is to do the innermost ones first. So let’s start solving this equation by breaking it down.

The (D*B) and the (F-7) are the innermost parts of the equation.

Let’s start with the (D*B).

The D * B = 4 * 2 = 8,

so we simplify the equation to,

(8 + (F-7) + A) * C = X

Next is the (F-7).

F - 7 = 6 - 7.

This results in a number one less than zero, so we say negative one or -1.

(Another example would be 15-20. This results in a number 5 less than zero, so we say negative 5 or -5.)

The equation now looks like,

(8 + (-1) + A) * C = X

Let’s get the A and C taken care of; the equation is now,

(8 + (-1) + 1) * 3 = X

Net we add up the numbers inside the parenthesis.

-1 plus 1 equals zero of course.

[Or you could have said: -1 plus 8 equals 7. The 8 is called a positive number, just as the 1 is called a negative number. Adding a positive number to a negative number is really just subtracting the negative number from the positive number. In other words:

8 + (-1) = 8 - 1 = 7 or 1 + (-1) = 1 -1 = 0 ]

Either way, our equation now looks like,

(8 - 1 + 1) * 3 = X, which is

(8) * 3 = 24 = X, or

8 * 3 = 24 = X, or

X = 24

Simplified a step at a time and solved.

If you didn’t know negative numbers before, then congratulations! Now you do. For the sake of completeness, the next section is about what else you should know about negative numbers.

More Example NASA Formulas

In Algebra How Do You Solve for V? Learning and doing volume formulas.
In Algebra How Do You Solve for V? Learning and doing volume formulas.

Negative Numbers

Numbers plus negative numbers result in lesser numbers. Keep in mind -10 is a lesser number than -5, etc.

Numbers minus negative numbers result in larger numbers. For example, whereas 9-5 = 4, but 9-(-5) = 14. In other words, minus minus results in a positive increase a.k.a a lesser lesser or a larger larger. Minus a minus is exactly the same as plus a plus, e.g. -(-25)=25.

This is a good time to mention that in mathematics, two negatives equal a positive when applied to minus a minus subtraction, or any multiplication, or any division.

For multiplication:

  • Negative numbers times positive numbers equal negative numbers, e.g. -5 * 4 = -20.
  • Negative numbers times negative numbers equal positive numbers, e.g. -5 * -4 = 20.
  • You already knew positive numbers times positive numbers equal positive numbers.

For division, the same rules apply:

  • Negative numbers divided by positive numbers (or vice versa) equal negative numbers, e.g. -5/4 = -1.25 and 5/-4 = -1.25.
  • Negative numbers divided by negative numbers equal positive numbers, e.g. -5/-4 = 1.25.

You already knew positive numbers divided by positive numbers equal positive numbers.

Using Spreadsheets

Spreadsheet software or applications will happily do the arithmetic and sort out the negatives versus the positives for you once you have replaced all the variables. It even knows to do the innermost before the outermost, etc. As an example, suppose you have simplified an equation to the following mess:

X=((5-3)* 52)-21+((6+7)/(34-12))

If your spreadsheet software is MS Excel or you are using cloud Google Drive, you can exclude the X and just copy/paste the following into a single cell:

=((5-3)* 52)-21+((6+7)/(34-12))

The spreadsheet will immediately solve the equation and give back the answer of 83.5bunchmoredigits. If you have the software or Google Drive access, go ahead and try it.

If you are really good at spreadsheet calculations, you can, of course, do equations with the variables still in place; substituting the variables with cell locations or range names.

Another Division Example

Time to include division. Might as well keep it simple and use the previous example.

A = 5

B = 34

C = 21

X=((A-3)* 52)-C+((6+7)/(B-12))

We replace the variables with the assigned numbers and we are right back where we started from:

X=((5-3)* 52)-21+((6+7)/(34-12))

The arithmetic then gives us:

X = 83.59090909…

Dividing by Zero

This is a good time to mention you cannot divide by zero.

For example

A=1

A=2

A=3

X = 5 + 10/(3-A)

Now if A=1, then

X=5+10/(3-1)=5+10/2=5+5=10

Now if A=2, then

X=5+10/(3-2)=5+10/1=5+10=15

If, however, we attempt to declare the variable A as A=3, the following occurs:

X=5+10/(3-3)=5+10/0. (invalid)

At this point the equation becomes invalid. There is no answer to the question, “What is 10 divided by 0?”. An equation immediately becomes invalid when a divide-by-zero scenario occurs. Software applications are designed to recognize this when it happens. Plugging whatever-divided-by-zero into a spreadsheet used to give interesting results, before applications were modified to detect this.

Show Off What You've Learned

Final Example

The basic concept of algebra is just plugging the numbers into the variables, and then doing the arithmetic. One merely keeps simplifying the equation until it is solved. You now have a full understanding of that concept. Yes, you have been using variables since the first paragraph.

Here is the last example. It is presented in a different format. The question, however, remains the same. What is X? You already know everything needed to solve this equation.

A=1, B=2, C=3, D=4, E=5

T=-1, U=-2, V=-3

(6X/8)+(2T+4)=((CD/2)-AD)+V

It should be noted 6X means the same as 6*X; and AD means the same as A*D. Other examples would be: 3A=3*A=A*3, 5Y=5*Y=Y*5, -2C=-2*C=C*-2, etc.

We plug in the variables, and the equation now is:

(6X/8)+((2*-1)+4)=((3*4)/2)-(1*4)+-3

Some simplifying arithmetic gives us:

(6X/8)+-2+4=(12/2)-4+-3

More arithmetic gives us:

6X/8 +2=6-4+-3

More arithmetic gives us:

6X/8+2=-1

We can’t solve X as the equation is currently stated; so we will have to move things around and do more arithmetic.

Important Note

Whenever you change the actual value on one side of the equation; you must do the same on the other side of the equation. Example: 7=7; if you subtract 2 from the left side, then you must subtract 2 from the right side.; 5=5. The same rule applies for addition, multiplication, and division.

Let’s subtract 2 from both sides of our equation.

6X/8+2=-1

Then becomes:

6X/8=-3

We have to get rid of the “divide by 8” part of the left side of the equation. So we multiply both sides of the equation by 8.

6X/8=-3

Then becomes:

6X=-24

We must make the X stand alone, so we divide both sides by 6.

6X=-24

Then becomes:

X = -4 (The Answer!)

How Do We Know If We Have the Right Answer?

To find out, we go back to the original equation and replace X with -4. We then simplify (reduce) the equation as before to its simplest form. If the simplest possible construct is valid; then, by definition, the statement “X=-4” is valid.

Here is the original equation.

A=1, B=2, C=3, D=4, E=5

T=-1, U=-2, V=-3

(6X/8)+(2T+4)=((CD/2)-AD)+V

We don’t have to re-solve the parts that didn’t have the X in it to begin with, so we have:

(6X/8)+2 = -1

We now replace the X with -4, giving us:

((6*-4)/8)+2=-1

Simplifying gives:

(-24/8)+2=-1

Which is:

-3+2=-1

Which is:

-1=-1

This construct is valid and simple enough to know X=-4 is valid.

To take it to the very end, you can multiply both sides by -1, giving you:

1=1

What Would Have Happened, If Instead of Correctly Calculating X=-4, We Had Erroneously Calculated X=16?

The equation simplification / reduction would have proceeded smoothly to this point:

(6X/8)+2 = -1 (as above)

When the 6X is replaced with 6*16, we get:

(96/8)+2=-1 (false)

When further simplified says:

12+2=-1 (false)

Which is

14=-1 (false)

The resulting false statement by definition means that the X=16 calculation is a false statement.

The Adventure Continues...

There is a lot more (a very lot more) to algebra, but it is really only an expansion of what you have already learned. Algebra is the basis of all other mathematics; including geometry, trigonometry, calculus, and so on. A good understanding of algebra is required to succeed at the other mathematics. Mathematics, itself, is the foundation of most other disciplines. This foundation is not just necessary for the sciences such as physics, electronics, chemistry, biology, astronomy, and so on. A mathematical foundation is necessary for many careers; including marketing, economics, architecture, and many, many others.

May all your calculations be prosperous ones!

Questions & Answers

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      • DzyMsLizzy profile image

        Liz Elias 

        4 months ago from Oakley, CA

        WHEW!! My brain hurts! LOL

        I tried and tried and tried to 'get' algebra. Took it in high school, didn't get it; dropped the class because I didn't want an "F" on my transcript.

        Tried again as an adult in community college. Monopolized the teacher's office hours; became well-known in the learning center; hired a private tutor. I still couldn't grasp it.

        It seems so arbitrary. Your explanations make the very basics of addition and subtraction clearer, but I still am flummoxed by the division and multiplication. (I suck at basic math in the first place.)

        But the thing that really gets me is HOW IN BLAZES do you end up with a positive number when you subtract 9-(-5) comes out to 14, when, according to everything that makes any kind of sense to ME, it should be -14.

        I understand negative numbers only when it affects my checkbook. (LOL)

        I understand in English grammar, how a double negative negates the statement, for instance, "She doesn't have no sugar," ends up meaning she does have sugar.

        But, in my mind, it just doesn't seem to translate across to numbers, and make sense. That's where I always thought it seemed arbitrary, and why I fall flat on my face every time; because I'm trying to make sense of it, and I can't.

        That, and each new problem looks so different that I can't seem to apply the formula, becuase it isn't parsed (English grammar term; don't know if it applies to math), just like the example problem(s).

        Oh, and BTW---WHY is algebra a pre-requisite to geometry???? Geometry was invented first!!!

        I'll stick to writing, methinks. ;-)

      • profile image

        Meriwether Clark 

        5 months ago

        Thank you so much!!!

      • profile image

        John r. Hunt 

        6 months ago

        great job

      • profile image

        Ashley Anne Hernandez 

        9 months ago

        Thanks a lot !!!

      • profile image

        Amanda 

        11 months ago

        I was always bad in math especially algebra , this really helps, but my question is , is there a book I can buy using these symbols ?

      • profile image

        I dont like math 

        13 months ago

        so glad I found this page!! I did 5 line papers with this and I hope my mom is proud when she sees all the work I did. :)

      • profile image

        dorothy jordan 

        2 years ago

        Im so thankful for all the help an support,i have a big test coming up,was so worried,but now ,i feel so secure,thanks for ur help

      • Kristen Howe profile image

        Kristen Howe 

        3 years ago from Northeast Ohio

        Math was always a weak subject for me. This would be helpful and useful for anyone who's struggling with it. Voted up for useful!

      • Jean Bakula profile image

        Jean Bakula 

        4 years ago from New Jersey

        Where were you when I was struggling with Algebra in 7th grade? And you even have pictures. I tried so hard, I used to stay after school for extra help. Finally the teacher agreed to pass me with a D if I would stop coming. I thought it was a good deal, although I was an A student in other subjects, I just couldn't grasp it. My son is now a teacher, and tells me she was a failure as a teacher for not trying harder!

      • profile image

        abseli 

        5 years ago

        Hi there! I could have sworn I've been to this website before but after browsing through some of the post I realized it's new to me. Anyways, I'm definitely delighted I found it and I'll be bookmarking and checking back frequently! make your computer run like brand new

      • Marderius profile image

        Marderius 

        5 years ago from Alabama

        Well 2/3 x 5 7/8 ..first we can change 5 7/8 into an improper fraction giving us 47/8 now we can cross multiply... we will have 2/3 x 47/8= as you can see 2 can go into itself one time and into 8 four times.... now we have 1/3 x 47/4 giving us a final answer of 47/12 or 3 11/12 ! Hope this was helpful

      • profile image

        Carole Garfield 

        5 years ago

        I helping someone in fraction like 2/3 x 5 7/8 = Can someone take me through this

      • carlarmes profile image

        carlarmes 

        6 years ago from Bournemouth, England

        my son found this hub useful, thank you for the content.

      • Angela_1973 profile image

        Angela_1973 

        7 years ago

        Good hub on algebra, I hate it though, I speak like 5 languages but can never do math. Have a lot of respect for people in the science field

      • kathryn1000 profile image

        kathryn1000 

        7 years ago from London

        That is really good.Congratulations.

        *Algebra helps your brain*

        To forget about troubles and pain.

        So dive in and swim

        While wearing a grin,

        And soon you'll be back here again

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