Find practice problems and solutions for these concepts at The Basics of Algebra Practice Problems.
What's Around The Bend
 Variables
 Expressions
 Coefficients
 Like Terms
 Simplifying Expressions
 The Commutative Property
 The Associative Property
 The Identity Element
 Additive Inverse
 Multiplicative Inverse
 The Distributive Property
Just when you were getting used to numbers, algebra attacks. You were hip to decimals, multiplication, even fractions. But algebra is a different kind of animal. It contains letters! Yes, letters. And that's not the only thing that makes algebra stand out.
You may be asking why you need to study algebra. Well, all the math that is needed in science and engineering relies on algebra as its basic language. Do you like toasters and TVs? Do you like airplanes and air conditioning? Do you like all the smart, edgy stuff that technology makes available? It all depends on algebra.
Letters in Math Class?
Variables are letters that are used to represent numbers. Once you realize that these variables are just numbers in disguise, you'll understand that they must obey all the rules of mathematics, just like the numbers that aren't disguised. This can help you figure out what number the variable at hand stands for.
Fuel for Thought
An expression is like a series of words without a verb. Take, for example, 3x + 5 or a – 3. A verb, which would be an equality or inequality symbol, would give value to the statement, turning it into an equation or inequality. These symbols include =, ≠, >, <, ≥, and ≤.

Look at this equation:
a + 4 = 7
In the equation a + 4 = 7, the letter a represents a little box. Think of the letter a as the label on the box. Inside the box is a number. Your job is to figure out what that number is. What number can you put in the box to turn the equation a + 4 = 7 into a true statement?
Let's just stick 20 into the a box. The equation becomes 20 + 4 = 7, or 24 = 7. Okay, this equation is not a true statement because 24 does not equal 7. So, a is not the number 20.
If you put 3 into the box, the equation becomes this true statement: 3 + 4 = 7, or 7 = 7.
Fuel for Thought
When a number is placed next to a variable, indicating multiplication, the number is said to be the coefficient of the variable. For example,
8c 8 is the coefficient of the variable c.
6ab 6 is the coefficient of both variables, a and b.
If two or more terms have exactly the same variable(s), they are said to be like terms.
7x + 3x = 10x
The process of grouping like terms together and performing their mathematical operations is called combining like terms. It is important to combine like terms carefully, making sure that the variables are exactly the same.

Simplifying Expressions
In order to simplify expressions, there are several things you have to know.
To simplify a – (b – c), just say no to the subtraction. Convert the expression to a + –1(b – c), which then becomes a + –b + c or a – b + c. Of course, you can go directly from a – (b – c) to a – b + c once you get the hang of it.
Inside Track
Don't forget the rules for order of operations with numerical expressions. You can use a memory device called a mnemonic to help you remember a set of instructions. Try remembering the acronym PEMDAS. This will help you remember to
P 
do operations inside Parentheses 
E 
E evaluate terms with Exponents 
M D 
do Multiplication and Division in order from left to right 
A S 
Add and Subtract terms in order from left to right 

When simplifying expressions, you should also start working as deeply inside the parentheses or braces as you can.
3 – (4 – [5 – 2(a – b)]) =
3 – [4 – (5 – 2a + 2b)] =
3 – (4 – 5 + 2a – 2b) =
3 – [–1 + 2a – 2b] =
3 + 1 – 2a + 2b =
4 – 2a + 2b
Solving for the Variable
When you want to solve for a variable in an algebraic equation, you want to get your variable all by itself. We call this "isolating" the variable. In order to preserve the equality of the given equation, you need to be sure that you are doing the same thing to both sides of the equation. This means that you should perform corresponding operations on both sides of the equal sign. If you subtract 2 from the left side, you need to subtract 2 from the right side. If you divide the left side by 3, you must divide the right side by 3.
Example
Evaluate the expression 2b + a when a = 2 and b = 4.
Substitute 2 for the variable a and 4 for the variable b. When the expression is written as 2b, it means 2 times b: 2(4) + 2. Multiply 2 · 4. Finally, add 8 + 2 = 10.
Caution!
When two or more numbers or variables are being multiplied, they are called factors. The answer that results is called the product.
5 × 6 = 30 
5 and 6 are factors, and 30 is the product. 
When you are working with variables, it may be easy to confuse the variable x with the multiplication sign ×. To avoid this, use the symbol · to represent the multiplication sign.
Multiplication is also indicated when a number is placed next to a variable. For example, look at 5a = 30. In this equation, 5 is being multiplied by a.
Also, understand that parentheses around any part of one or more factors indicates multiplication: (5)6 = 30, 5(6) = 30, and (5)(6) = 30.

Algebra is governed by certain axioms. Axioms are statements that you regard as true because they are obvious and because your knowledge has to start somewhere. Other statements you need, beyond the axioms, can be proven based on the axioms.
The Basic Axioms
In this chapter, you will discover a basic set of axioms. Four axioms have to do with addition, and four axioms have to do with multiplication. The ninth axiom brings addition and multiplication together. The following chart offers a brief summary of these axioms.
The Commutative Property
The commutative property allows you to change the order of the numbers when you add or multiply.
Let's define addition, a + b, as repeated counting, so that 3 + 5 means to start at 3 and count five times: 4, 5, 6, 7, 8. Then, 5 + 3 means to start at 5 and count three times: 6, 7, 8. You can see that adding will come out to the same result both ways.
For multiplication, define 3 · 5 as repeated adding: 3 times 5 means to start with zero and add 5 three times: 0, 5, 10, 15. And 5 times 3 means to start with zero and add 3 five times: 0, 3, 6, 9, 12, 15. Remarkably, the result will always come out the same both ways.
And then, after seeing that the commutative property of addition and of multiplication makes sense for integers, you can make the leap and accept these laws for any numbers, whether they are integers or not.
The Associative Property
Adding is an operation you perform on two numbers at a time. So, the expression 3 + 4 + 5 has to mean either:
(3 + 4) + 5 
(3 + 4, which is 7, added to 5) 
or 

3 + (4 + 5) 
(the number 3 added to 4 + 5, which is 9) 
In the first expression, the 4 "associates" with the 3; in the second expression, the 4 associates with the 5.
The associative property of addition guarantees that both expressions will yield the same number, regardless of which numbers you add first. In the first example, the result is 7 + 5 = 12; in the second example, 3 + 9 = 12. The same result, 12, occurs in both cases.
In the case of multiplication, 3 · 4 · 5 can be interpreted either as
(3 · 4) · 5 
(3 · 4, which is 12, multiplied by 5) 
or 

3 · (4 · 5) 
(the number 3 multiplied by 4 · 5, which is 20) 
The associative property of multiplication asserts that the two examples will result in the same number: 12 · 5 = 60 and 3 · 20 = 60.
The Identity Element
Zero is the leaveitalone number of addition—more formally known as the identity element of addition.
Adding zero to 17 results in 17: 17 + 0 = 17. Expressed differently, adding zero to 17 preserves the identity of 17: 17 + 0 = 17.
One is the leaveitalone number of multiplication. Multiplying 17 by 1 leaves the 17 alone: 17 · 1 = 17. Expressed differently, multiplying 17 by 1 preserves the identity of 17: 17 · 1 = 17.
Pace Yourself
When you buy a new or used car, you may have to pay sales tax in addition to the price of the car. Go to a car lot or browse the newspaper for a vehicle that you would be interested in buying. If the sales tax is 5%, calculate the sales tax on the vehicle and then determine the total cost of the car. Begin by setting up an equation.

The Additive and the Multiplicative Inverses
The additive inverse, or opposite, of a number n is the number that, when added to n, equals zero. The additive inverse of n is denoted –n. Every number has an additive inverse. The opposite of 3 is –3, and the opposite of –3 is 3. These numbers satisfy 3 + –3 = 0.
If you take 17 and add 3 to get 20, you can get back to 17 by adding –3. And if you have 17 and you add –3 to get to 14, you can get back to 17 by adding 3. This is how the "oppositeness" of 3 and –3 works.
Multiplication works the same way. You get the multiplicative inverse by inverting the number. A number times its multiplicative inverse equals 1.
The reciprocal of 3 is , and the reciprocal of is 3. Three and are reciprocals of each other. These numbers satisfy 3 · = 1. Following are some other examples.
Examples
2 · = 1
· = 1
– · –5 = 1
If you have 17 and you multiply by 3 to get 51, you can get back to 17 by multiplying 51 by . And if you have 17 and multiply by to get , you can get back to 17 by multiplying by 3. This is how the reciprocal nature of 3 and works.
These "do" and "undo" properties will aid you when you solve equations. Suppose you have the equation 3x + 4 = 19.
Solving equations like this, which is a very important part of algebra, means to arrive at a statement that says, x = … (whatever it comes out to). In other words, the goal is to isolate the x.
On the lefthand side of this equation, x has been multiplied by 3, and 4 has been added. You can undo these operations by doing the inverse operations—adding –4 and multiplying by .
3x + 4 + –4 = 19 + –4
3x = 15
3x · = 15 ·
x = 5
That's how you can use inverses to your advantage in the world of algebra!
The Distributive Property
What do you do with a problem like this: 2(x + y) + 3(x + 2y)? According to the order of operations that you learned earlier in this chapter, you would have to do the symbols in parentheses first. However, you know you can't add x to y because they are not like terms. What you need to do is use the distributive property. The distributive property tells you to multiply the number and/or variable(s) outside the parentheses by every term inside the parentheses.
For this problem, multiply 2 by x and 2 by y. Then, multiply 3 by x and 3 by 2y. If there is no number in front of the variable, it is understood to be 1, so 2 times x means 2 times 1x. To multiply, you multiply the numbers and the variable stays the same. When you multiply 3 by 2y, you multiply 3 by 2 and the variable, y, stays the same, so you would get 6y. After you have multiplied, you can then combine like terms.
Fuel for Thought
If there is no number in front of a variable, it is understood to be 1.

Example
5(a + b) = 5a + 5b
This can be proven by doing the math:
5(1 + 2) = (5 × 1) + (5 × 2)
5(3) = 5 + 10
15 = 15
Find practice problems and solutions for these concepts at The Basics of Algebra Practice Problems.