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Chapter 6—Expressions and Equations

6-1 Introducing Expressions and Equations

Algebraic expressions and equations:  this is where your work really begins to look and feel like algebra. You are about to begin working with letters of the alphabet as well as numbers and signs of operation. From now on, letters such as x, y, and z are just as common as numbers 1, 2, and 3.

So what do letters in algebra mean? Until now –- in basic arithmetic –- you worked with numbers, each number having a specific value or meaning. A "2" is a 2, for example. A "2" is always a 2 –- it is never a 6 and it is never a 10. It's like that for all numbers.  We still use regular arithmetic numbers in algebra,  but we also use terms expressed in letters. In algebra, the letter x, for example, can stand for a lot of different values. We can set x equal to 2 in one problem, but then set it equal to 6 in another. The letters in algebra can stand for an endless variety of values and combinations of values. Letters in algebra can even represent other letters.

Compare these two expressions:

• Arithmetic expression:  2 + 1
• Algebraic expression: x + 1

The arithmetic expression tells us to add 1 to the value of 2. The algebraic expression, however, covers a lot more territory by telling us to add a 1 to any value we choose. So if we let x = 2 in the algebraic expression, it becomes 2 + 1. If we let x = 5, it becomes the same as 5 + 1. You can see that the algebraic version is a lot more flexible than the arithmetic version.

 Definitions A specific numerical value ( such as 2, 4, -6, ¾ ) is called a constant. The value is constant ... unchanging. An algebraic term (such as x, y, a, b, and so on) is called a variable. The value can be varied. When constants or variables are connected by operations (such as +, -, x , or ÷ ), you have an expression

Examples of Algebraic Expressions

These slides give you some common examples of algebraic expressions.

When you are done with this set of slides, you should be able to speak the expression; describe exactly what it means; and  identify the variables, constants, and sign of operation.

 Definition An equation is a statement of equality between two expressions. It consists of two sets of algebraic expressions separated by an equal sign 5 + y = 20 Equation

The purpose of an equation is to express equality between the two expressions. And what is the real difference between an algebraic expression and an algebraic equations? Simple: An equation includes an equal sign (=) and an expression does not. An expression can include signs of operation, but not an equal sign.

Note: The main difference between an expression and an equation is that an equation includes an equal sign. An expression does not.

Examples of Algebraic Equations

These slides give you some common examples of algebraic equations.

When you are done with this set of slides, you should be able to "say" the equation,  describe exactly what it means; and  identify the variables, constants, and sign of operation.

### What is This All  About? A clear, workable understanding of the world is not depend on knowing a zillion little facts. There are people who have memorized entire encyclopedias, the Bible, and the Guinness Book of Records, but that does not give them any special advantage in dealing with reality.  A clear, workable understanding of the world depends much more on our ability to think,  learn and "see" principles that are not apparent to most people.

Your own path in  life might never require you to simplify a lot of algebraic equations, but the kind of thinking, analysis, and insight you develop through lessons such as these, will serve you well in many challenging situation -- most  having nothing to do  with math, but requiring the same level of analytical skills.

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