Basic Concepts: Variables and Equations

1. Algebra is simply a logical extension of arithmetic. The same four fundamental operations you learned in arithmetic are also essential in algebra: addition (+), subraction (-) multiplication (X), and division (÷). The result of addition is the sum; of subtraction, the difference, or remainder; of multiplication, the product, and of division, the quotient.

2. Algebra differs from arithmetic in its frequent use of letters to represent numbers.

Arithmetic: 2 + 3 = 5 Algebra: a + b = c

2. The four operations are performed in algebra with one major difference. In algebra letters frequently are used to represent numbers. Why? Because in algebra we often work with quantities without regard to their numerical values. We may need to use their numerical values eventually, but in the meantime we have to identify them in some way. So we use letters of the alphabet.

3. The use of letters to represent numbers makes it possible to translate long word statements into short mathematical sentences, expressions, or statements.

Word statement: The difference between twice a number (n) and half that number is nine. Mathematical statement:

3. How does the use of letters, numbers, and symbols make it possible to translate long word statements into brief mathematical statements? Here is an example:

The sum of five times a number and two times the same number is equal to seven times the number. How can we represent this more simply?

8n - 3n = 5n

Use letters and symbolc to change these word statements into algebraic expressions:

(a) The sum of one-half x and one x equals 12.

(b) Twice d plus half of b added to 3 equals nine.

(c) Ten times a number (n) minus three times the number equals 7 more than four times the number.

(d) The area (A) of a triangle is equal to one-half of the base (b) times the height (h).

4. A letter used to represent a number is called a literal number or variable.

In the equation t + 3 = 7, the letter t is a literal number or variable.

4. The word literal means having to do with a letter (of the alphabet). In algebra, we have a psacial name for a letter that is used to represent a number. It is called a literal number or variable.

5. An algebraic statement that represents two things that are equal to one another is called and equation.

8n - 3n=5n

5. The word statements which you translated into an algebraic expression in item 3 above are examples of equations since, in each case, one quantity was equal to another. Bear in mind that an algebraic expression is not necessarily an equation, unless there is an equality involved. For example, ax + by + c is an an algebraic expression; ax + by + c = 0 is an algebraic expression in the form of an equation. An equation will always contain and equal sign (=).

Next Session: Symbols, Factors, Coefficients and Axioms