# A Math Primer

## Algebraic Expressions

**Variables**

Letters represent an unknown or generic real

number

Sometimes with restrictions, such as a member

of a certain set, or the set of values that makes an

equation true.

Often a letter from the end of the alphabet: x, y, z

Or a letter that stands for a physical quantity: d

for distance, t for time, etc.

**Constants**

Fixed values, like 2 or 7

Can also be represented by letters: a, b, c, p, e, k

**Terms**

Terms are Separated by + or –

**Factors**

Factors are multiplied together.

**Coefficients**

Coefficients are constant factors that multiply a

variable or powers of a variable

The middle term has 2 factors, –3 and x. We say that

the coefficient of x is –3.

The first term has three factors, 2 and two factors of

x. We say that 2 is the coefficient of x^{2}.

The last term is a factor all by itself (although the

number 4 could be factored into 2 x 2).

**Simplifying Algebraic Expressions
**

By “simplifying” an algebraic expression, we mean

writing it in the most compact or efficient manner,

without changing the value of the expression. This

mainly involves collecting like terms, which means

that we add together anything that can be added

together. The rule here is that only like terms can be

added together.

**Like (or similar) terms**

Like terms are those terms which contain the same

powers of same variables. They can have different

coefficients, but that is the only difference.

**Examples:**

3x, x, and –2x are like terms.

2x

^{2}, –5x

^{2}, and are like terms.

xy

^{2}, 3y

^{2}x, and 3xy

^{2}are like terms.

xy

^{2}and x

^{2}y are

**NOT**like terms, because the same

variable is not raised to the same power.

**Combining Like terms**

Combining like terms is permitted because of the

distributive law. For example,

3x

^{2}+ 5x

^{2}= (3 + 5)x

^{2}= 8x

^{2 }

What happened here is that the distributive law was

used in reverse—we “undistributed” a common

factor of x

^{2}from each term. The way to think about

this operation is that if you have three x-squareds,

and then you get five more x-squareds, you will then

have eight x-squareds.

**Example:**x

^{2}+ 2x + 3x

^{2}+ 2 + 4x + 7

Starting with the highest power of x, we see that

there are four x-squareds in all (1x

^{2}+ 3x

^{2}). Then we

collect the first powers of x, and see that there are six

of them (2x + 4x). The only thing left is the constants

2 + 7 = 9. Putting this all together we get

x

^{2}+ 2x + 3x

^{2}+ 2 + 4x + 7

=

4x

^{2}+ 6x + 9

**Parentheses**

∞ Parentheses must be multiplied out before

collecting like terms

You cannot combine things in parentheses (or other

grouping symbols) with things outside the

parentheses. Think of parentheses as opaque—the

stuff inside the parentheses can’t “see” the stuff

outside the parentheses. If there is some factor

multiplying the parentheses, then the only way to get

rid of the parentheses is to multiply using the

distributive law.

**Example:**

3x + 2(x – 4) = 3x + 2x – 8

= 5x – 8

**Minus Signs: Subtraction and Negatives
**

Subtraction can be replaced by adding the opposite

3x – 2 = 3x + (–2)

**Negative signs in front of parentheses**

A special case is when a minus sign appears in front

of parentheses. At first glance, it looks as though

there is no factor multiplying the parentheses, and

you may be tempted to just remove the parentheses.

What you need to remember is that the minus sign

indicating subtraction should always be thought of as

adding the opposite. This means that you want to add

the opposite of the entire thing inside the

parentheses, and so you have to change the sign of

each term in the parentheses. Another way of looking

at it is to imagine an implied factor of one in front of

the parentheses. Then the minus sign makes that

factor into a negative one, which can be multiplied

by the distributive law:

3x – (2 – x)

= 3x + (–1)[2 + (–x)]

= 3x + (–1)(2) + (–1)(–x)

= 3x – 2 + x

= 4x – 2

However, if there is only a plus sign in front of the

parentheses, then you can simply erase the

parentheses:

3x + (2 – x)

= 3x + 2 – x

**A comment about subtraction and minus signs
**Although you can always explicitly replace

subtraction with adding the opposite, as in this

previous example, it is often tedious and

inconvenient to do so. Once you get used to thinking

that way, it is no longer necessary to actually write it

that way. It is helpful to always think of minus signs

as being “stuck” to the term directly to their right.

That way, as you rearrange terms, collect like terms,

and clear parentheses, the “adding the opposite”

business will be taken care of because the minus

signs will go with whatever was to their right. If what

is immediately to the right of a minus sign happens

to be a parenthesis, and then the minus sign attacks

every term inside the parentheses.

**Solutions of Algebraic Equations**

Up until now, we have just been talking about

manipulating algebraic expressions. Now it is time to

talk about equations. An expression is just a

statement like

2x + 3

This expression might be equal to any number,

depending on the choice of x. For example, if x = 3

then the value of this expression is 9. But if we are

writing an equation, then we are making a statement

about its value. We might say

2x + 3 = 7

A mathematical equation is either true or false. This

equation, 2x + 3 = 7, might be true or it might be

false; it depends on the value chosen for x. We call

such equations conditional, because their truth

depends on choosing the correct value for x. If I

choose x = 3, then the equation is clearly false

because 2(3) + 3 = 9, not 7. In fact, it is only true if I

choose x = 2. Any other value for x produces a false

equation. We say that x = 2 is the solution of this

equation.

**Solutions**

The solution of an equation is the value(s) of the

variable(s) that make the equation a true

statement.

An equation like 2x + 3 = 7 is a simple type called a

linear equation in one variable. These will always

have one solution, no solutions, or an infinite number

of solutions. There are other types of equations,

however, that can have several solutions. For

example, the equation

x

^{2}= 9

is satisfied by both 3 and –3, and so it has two

solutions.

**One Solution**

This is the normal case, as in our example where the

equation 2x + 3 = 7 had exactly one solution, namely

x = 2. The other two cases, no solution and an infinite

number of solutions, are the oddball cases that you

don’t expect to run into very often. Nevertheless, it is

important to know that they can happen in case you

do encounter one of these situations.

**Infinite Number of Solutions**

Consider the equation

x = x

This equation is obviously true for every possible

value of x. This is, of course, a ridiculously simple

example, but it makes the point. Equations that have

this property are called identities. Some examples of

identities would be

2x = x + x

3 = 3

(x – 2)(x + 2) = x

^{2}– 4

All of these equations are true for any value of x.

The second example, 3 = 3, is interesting because it

does not even contain an x, so obviously its

truthfulness cannot depend on the value of x! When

you are attempting to solve an equation algebraically

and you end up with an obvious identity (like 3 = 3),

then you know that the original equation must also be

an identity, and therefore it has an infinite number of

solutions.