Chapter 8    Expressions and Equations

8-1    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 work 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 expressions 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 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.

 Expression Spoken Meaning Significant Parts Notes x + 2 "x plus two" Add variable x and constant 2 x is a variable 2 is a constant + is the sign of operation This is an example of an algebraic expression that consists of one variable and one constant. y – 5 "y minus five" Subtract constant 5 from variable y y is a variable 5 is a constant – is the sign of operation 5 – y "five minus y" Subtract variable y  from constant 5 y is a variable 5 is a constant – is the sign of operation 2x "Two times x" Multiply constant 2 by variable x x is a variable 2 is a constant No sign of operation indicates multiplication 2x can also be indicated by (2)(x) or 2 · x. The x multiplication sign should not be used in algebraic expressions because it is too easily confused with the variable letter x. w/4 "w divided by 4" Divide variable w by constant 4 w is a variable 4 is a constant / is the sign of operation 4/w "4 divided by w" Divide constant 4 by variable w w is a variable 4 is a constant / is the sign of operation x + y "x plus y" Add variables x and y x and y are variables + is the sign of operation This is an example of an algebraic expression that consists of two variables and no constants. x – 2y "x minus two times y" Multiply y by 2, and then subtract the result from x.Remember that the rules for order of operation call for doing multiplication before doing addition. x and y are variables 2 is a constant – and the "no sign" of multiplication are the signs of operation This is an example of an algebraic expression that consists of two variables and one constant.

 Definition An equation is a statement of equality between two expressions.It consists of two sets of algebraic expressions separated by an equal sign

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 the presence of an equal sign in one of them. Which one?

Examples of algebraic equations.

 Equation Spoken Meaning Significant Parts Notes x + 3 = 5 "x plus 3 equals 5" Expression x + 3 is equal to the constant value 5 x is a variable 3 and 5 are constants + is a sign of operation x – 12 = 16 "x minus 12 equals 16" Expression x – 12 is equal to the constant value 16 x is a variable 12 and 16 are constants – is a sign of operation x = y+ 2 "x equals y plus 2" Variable x is equal to the expression y + 2 x and y are variables 2 is a constant + is a sign of operation This is an example of an equation that has two variables, one constant, and one sign of operation.. x = 2y "x equals 2y" The value of x is twice the value of y. x and y are variables 2 is a constant The sign of operation is understood to be multiplication. x/2 + 4 = 3y – 8 "x divided by 2 plus 4 is equal to 3 times y munus 8" The value of expression x/2 + 4 is equal to the expression 3y – 8. x and y are variables 2, 4, and 3 are constants The sign of operation include division, addition, multiplication, and subtraction. This is an example of an equation that has two variables, three constants, and four signs of operation.