8th Grade Algebra Review - The Language  

Algebra is the branch of mathematics that is a generalization of arithmetic; it uses letters, symbols, and/or characters to represent numbers and express mathematical relationships. Those symbols are called variables.

Variables are symbols used to represent unspecified numbers or values. A letter, such as x, y s, or t may be used as a variable.

An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations. Here are some examples:

4y means 4 times the variable y

2a - 6 means 6 subtracted from 2 times the variable a

x · y means the variable x multiplied by the variable y

(x)(y) also means the variable a multiplied by the variable y, as does xy and x(y) and (x)y

In these expressions the quantities being multiplied are called factors and the result is called the product.

An expression like a/b is called a quotient.  It is one varible divided by another.

An expression like yⁿ is read y the the nth power. The variable y is called the base, and n is called the exponent. I meqns we multiply the variable v by itself n times. So  y³ means y multiplied by itself three times, or y · y · y

An algebraic expression can consist of one or more terms. A term is a number, a variable, a product of numbers and variables, or a quotient of numbers and variables. Examples:

5  this is a term with one number.

x²    this is a term with a variable

7y   this is a term that is a product of a number and a variable

3x/y   this is a term that is a quotient of number and a variable and another variable.

The coefficient of a term is the numerical factor of the term. The coefficient of 25mn is 25.

Two expressions are equivalent expressions if they denote the same number. For example:

3x + 12x  and 15x are equivalent expressions.

An expression is in its simplest form when it is replaced by an equivalent expression having no like terms or parentheses.

The simplest form of: 12a + (3 +7)a is:

12a + (10)a = 12a + 10a = 22a

An operation is an arithmetic operation such as +, -, · , or ÷. Another operation is raising a number or variable to a power.

In algebra, we perform arithmetic operations according to a rule called the order of operations. Here is the order:

1. Evaluate expressions inside grouping symbols, such as

( 4 + 7). Evaluate the inner most expressions first.

2. Evaluate all powers.

3. Do all multiplications and/or divisions from left to right.

4. Do all additions and subtractions from left to right.

Example 1:

24 ÷ 6 · 5 - (4 -2)^{3 =}
 

24 ÷ 6 · 5 - (2)³   Step 1

24 ÷ 6 · 5 - 8    Step 2

20 - 8    Step 3

12   Step 4

Example 2:

2[ 6 + (30 ÷ 5)² ]

2[ 6 + (6)² ]  Step 1 evaluate inner most expression first

2[ 6 + 36 ]   Step 2 Evaluate the powers of the inner most expressions.

 2[ 42 ]  Step 3 evaluate the inner expression

84    Step 4 evaluate the multiplication.

An open sentence is a mathematical statement with one or more variables.

A sentence that contains an equal sign, =, is called an equation.

A sentence that contains the symbol <, ≤ , >,  or ≥ is called an inequality.

A set is a collection of objects or numbers.

A replacement set is a set of numbers from which replacements for a variable in an expression may be chosen. A solution set of an open sentence is the set of elements from the replacement set that make the open sentence true. Sometimes there is only one element in the solution set. For example, the solution set for: 5a + 8 = 48 is {8}.