Introduction

In this chapter, we are going to explore the foundational properties of Algebra. We will also explore the rules of addition and subtraction, the two most basic mathematical operations.

In arithmetic, we learned how to add, subtract, multiply, and divide numbers. Thus we know that statements such as

3 + 4 = 7

12 - 3 = 9

5 ⋅ 2 = 10

Are all true. The "⋅" symbol is the multiply symbol ("×" and "⋅" are interchangeable). Mathematical statements that contain the equal sign are also called equations.

On the other hand, these statements are false:

3 + 6 = 7

2 - 3 = 9

5 ⋅ 2 = 4

They are false because the left side of the equal sign (=) is not equal to the right side.

To demonstrate our knowledge, we solved problems such as the following:

7 + 9 = ___

12 ⋅ 3 = ___

In these problems, we evaluated the operation on the left side of = and wrote the result on the right side of =.

For example, since 7 + 9 is 16, the first blank is 16. And 12 ⋅ 3 is 36, so the second blank is 36. In other words:

7 + 9 = 16

12 ⋅ 3 = 36

Solving the problem led to a true statement.

We can also switch the left and right sides. Here are some examples:

___ = 2 + 5

___ = 11 + 1

Now we perform the calculations on the right side of = and write the result on the left side.

Next, let's try replacing operands with blanks. For example:

___ + 9 = 14

12 ⋅ ___ = 24

Can you fill in the blanks?

Since 5 + 9 = 14, the first blank is 5.

And since 12 ⋅ 2 = 24 the second blank is 2.

Variables

Sometimes blanks are not expressive enough. For example, consider the following set of equations:

__ + 13 = __

7 - __ = 5

The first equation has 2 blanks. We don't have enough information to fill in the blanks. But suppose that the first blank in the first equation is equal to the blank in the second equation.

Then we can first find the value of the blank in the second equation and use it to fill in the blanks in the first equation.

It would be nice if the equations can show that two of the blanks are the same. The problem is that we only have one symbol, the blank, to describe unknowns. Thus, we can simply use more symbols. Let's replace the blanks in the first equation with a and b. Then the first equation becomes:

a + 13 = b

Since the first blank of this equation is equal to the blank in the second equation, the second equation becomes:

7 - a = 5

We can easily find a from the second equation. 7 - 2 is 5, so

a = 2

Then the first equation becomes:

2 + 13 = b

Now we know that b is 15.

Using symbols made the blanks much easier to track.

Here are some more equations:

x + 9 = 14

12 ⋅ y = 24

In the first example, x = 5. In the second, y = 2

Pretty simple, right? The symbols are called variables, since their values can vary depending on the equation. Throughout this course, we will study equations that contain variables and try to find their values. Thus far, examples were simple enough that we could guess their solutions. But guessing is not a good strategy for more complex equations. Consider the following example:

12 ⋅ x + 8 = x - 2

Can you try to guess what x is?

In this course, we will learn how to solve equations like the example above. Let's begin by exploring fundamental mathematical properties and proving simple theorems.

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