How to Learn / Do Algebra. Equations. Math Tutor / Homework / Problems / Calculator. * A Mathematics Lesson *

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If you already know arithmetic (including fractions and decimals), then you already know algebra. You just don’t know that you know yet. If you understand the answers in the following statements, proceed with this article; otherwise, do not.

(the “+” sign is used to replace the word “plus”)

(the “-“ sign is used to replace the word “minus”

(the “=” sign is used to replace the word “equals”)

5 + 7 = 12

14 - 12 = 2

4 times 8 = 32

50 divided by 10 = 5

50 divided by 40 = 1.25

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

3 + 1 = 4

So, if we say that 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

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.

A = 9

B = 4

X = A - B. What is X?

We plug in the numbers and we get:

X = 9 - 4

X = 5

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

(the “*” 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

(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

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 doing equations since the first paragraph.

Time to include multiplication.

A=1

B=2

C=3

D=4

X=A+B*C-D. What is X?

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

So the equation written 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 that X=A and A=X are mathematically equal.

An interesting side note: What you have been and are doing is just 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 more than the rest of us can. In other words, they can’t grasp the whole equation either. They just proceed from line to line, trusting that they did the previous line(s) correctly.

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 parenthesis. When that happens, the rule is to do the innermost ones first. So let’s start 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 that 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 seven. 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

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.

Numbers plus negative numbers result in lesser numbers. Keep in mind that -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 aka 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 that 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 that positive numbers divided by positive numbers equal positive numbers.

Spreadsheet software 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 reduced an equation to the following mess:

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

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

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

You would immediately get back the answer of 83.5bunchmoredigits. If you have the software, 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.

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…

This is a good time to mention that 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 designed to detect this.

The basic concept of algebra is just plugging the numbers into the variables; and then doing the arithmetic. 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 that 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 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 figure out X as the equation is currently stated; so we will have to move things around and do more arithmetic.

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!)

To find out, we go back to the original equation and replace X with -4. We then 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 redo 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

Which is:

(-24/8)+2=-1

Which is:

-3+2=-1

Which is:

-1=-1

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

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

1=1

The equation 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)

Which is:

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.

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 foundation of most other mathematics: including geometry, trigonometry, calculus, and so on. A good understanding of algebra is required to succeed at mathematics. Mathematics, itself, is the foundation of most other disciplines. This foundation is not just required for the sciences such as physics, chemistry, biology, astronomy, ad infinitum. A mathematical foundation is necessary for many careers: such as marketing, economics, architecture, and many, many more.

May all your calculations be prosperous ones!

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