# Expressions and Equations

In the English language there are phrases and there are sentences, as you know. A phrase is a string of words that expresses a thought but does not form a complete sentence. A sentence is a grammatically correct string of words that includes both a subject and a verb. This is a sentence. Phrase over here.

**Some Phrases**

1. "the fat cat"

2. "purple feathers"

3. "upon the mountain"

4. "the bald eagle"

**Some Sentences**

1. "The fat cat liked mice."

2. "The bird has purple feathers."

3. "The hermit lived upon the mountain."

4. "The bald eagle was disappointed to find that the Rogaine wasn't working."

Not every string of words is a phrase or sentence. Some strings of words are nonsense. Like a good majority of the poppycock that comes twaddling out of our claptraps.

**Some Nonsense**

1. "fish dog cat and the"

2. "like swim eat chocolate"

3. "so if but or to"

We apologize if any of these look suspiciously like the text messages you send on a daily basis. We also apologize if the non-phrase "like swim eat chocolate" ever is the heading on your future online dating profile.

In algebra, instead of phrases and sentences, we have expressions and equations. An **expression** is a string of mathematical symbols representing a quantity. An **equation** is a string of mathematical symbols stating the equality of two expressions. Therefore, an expression is like a phrase and an equation is like a sentence. Instead of subjects and verbs, we have constants and variables. Instead of punctuation, we have symbols. We could go on like this all day.

**Some Expressions**

1. *x*

2. *3y* + 2*x* - 45(6 - *x*)^{2}

3. 0

**Some Equations**

1. *x* = 2

2. *4x* = 20

3. *3x* + 10*y* - 12 = 0

Notice that expressions do not have = signs, while equations *must* have = signs. An equation is a statement that two expressions have the same value, and the equal sign is a necessity. "Equation" even starts with the same letters as "equal," which should be a good clue.

In order to be an expression or equation, the string of symbols must make sense. Catch you drift our?

**Some Nonsense**

1. *xy* - + - 34

2. + + +

3. *x* 2 ÷ ( ( (

The pieces of an expression separated by + and - signs are called **terms**. A term will be positive if it follows a + sign and negative if it follows a - sign. When we break an expression into terms, we need to be careful to keep the negative signs in front of terms that are being subtracted. We also need to be careful not to get our thumb in the way when we are striking it with our ball-peen hammer.

### Sample Problems

1. In the expression 3*x* + 2, the terms are 3*x* and 2.

2. In the expression 3*x* - 2 the terms are 3*x *and - 2. This fact is true because 3*x* - 2 means 3*x* + ( - 2). Don't forget that there actually is a plus sign in front of positive numbers, even if we can't always see them. We knew we never should have given them that cloak of invisibility.

3. The terms in the expression are 5*x*, 3*y*^{2}, and .

Expressions that are multiplied together are called **factors**. You will need to come to grips with this information—it is just one of the factors of life.

### Sample Problems

1. 3 and *x* are both factors of 3*x*.

2. In the expression (3 -* x*)(2*y*^{2} + 9), (3 - * x*) and (2*y*^{2} + 9) are both factors.

3. 5, * x*, and *y* are all factors of 5*xy*.