Introduction to Variables, Introduction to Variables, Algebraic Expressions, and Algebraic Expressions, and EquationsEquations

Section 1.8

A combination of operations on letters A combination of operations on letters (variables) and numbers is called an (variables) and numbers is called an algebraic algebraic expressionexpression..

Algebraic ExpressionsAlgebraic Expressions

5 + 5 + xx 6 6 yy 3 3 yy – 4 + – 4 + xx

44xx meansmeans 4 4 xx

andand

xyxy meansmeans xx yy

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Replacing a variable in an expression Replacing a variable in an expression by a number and then finding the by a number and then finding the value of the expression is called value of the expression is called evaluating the expression evaluating the expression for the for the variable.variable.

3Martin-Gay, Prealgebra, 5ed

Evaluate x + y for x = 5 and y = 2.

Evaluating Algebraic ExpressionsEvaluating Algebraic Expressions

x + y = ( ) + ( )

Replace x with 5 and y with 2 in x + y.

5 2

= 7

4Martin-Gay, Prealgebra, 5ed

EquationEquation

Statements like 5 Statements like 5 ++ 2 2 == 7 are called 7 are called equationsequations..

An equation is of the form An equation is of the form expression expression == expression expression

An equation can be labeled as An equation can be labeled as Equal sign

left side right side

x + 5 = 9

Solving/SolutionSolving/Solution

When an equation contains a variable, When an equation contains a variable, deciding which values of the variable deciding which values of the variable make an equation a true statement is make an equation a true statement is called called solvingsolving an equation for the an equation for the variable.variable.

A A solutionsolution of an equation is a value for of an equation is a value for the variable that makes an equation a the variable that makes an equation a true statement.true statement.

Martin-Gay, Prealgebra, 5ed

Solving/Solution ...Solving/Solution ...

Determine whether a number is a solution:Determine whether a number is a solution:

Is -2 a solution of the equation 2y + 1 = -3?

Replace y with -2 in the equation.

2y + 1 = -3

2(-2) + 1 = -3?

- 4 + 1 = -3

-3 = -3

?

TrueTrue

Since -3 = -3 is a true statement, -2 is a solution of the equation.

Solving/Solution ...Solving/Solution ...

Determine whether a number is a solution:Determine whether a number is a solution:

Is 6 a solution of the equation 5x - 1 = 30?

Replace x with 6 in the equation.

5x - 1 = 30

5(6) - 1 = 30?

30 - 1 = 30

29 = 30

?

FalseFalse

Since 29 = 30 is a false statement, 6 is not a solution of the equation.

To solve an equation, we will use To solve an equation, we will use properties of equality to write simpler properties of equality to write simpler equations, all equivalent to the original equations, all equivalent to the original equation, until the final equation has the equation, until the final equation has the form form xx == number number or or number number == xx

Equivalent equationsEquivalent equations have the have the samesame solutionsolution. . The word “number” above represents the The word “number” above represents the solution of the original equation.solution of the original equation.

Solving/Solution...Solving/Solution...

Keywords and phrases suggesting addition, subtraction, multiplication, division or equals.

AdditionAddition SubtractionSubtraction MultiplicationMultiplication DivisionDivision Equal Equal SignSign

sumsum differencedifference productproduct quotientquotient equalsequals

plusplus minusminus timestimes intointo givesgives

added toadded to less thanless than ofof perper isis//waswas// will bewill be

more thanmore than lessless twicetwice dividedivide yieldsyields

totaltotal decreased decreased byby

multiplymultiply divided divided byby

amounts amounts toto

increased increased byby

subtracted subtracted fromfrom

doubledouble is equal is equal toto

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the product of 5 and a numberthe product of 5 and a number55xx

twice a numbertwice a number22xx

a number decreased by 3a number decreased by 3nn -- 3 3

a number increased by 2a number increased by 2zz ++ 2 2

four times a numberfour times a number44ww

Translating Word Phrases Translating Word Phrases into Expressionsinto Expressions

11Martin-Gay, Prealgebra, 5ed

x ++ 7 7

three times the sum of a number and 7three times the sum of a number and 7

3(3(x ++ 7) 7)

the quotient of 5 and a numberthe quotient of 5 and a number

the sum of a number and 7the sum of a number and 7

5

x

Additional Word Phrases into Additional Word Phrases into Algebraic Expressions ...Algebraic Expressions ...

12Martin-Gay, Prealgebra, 5ed

Helpful HintHelpful Hint

Remember that order is important when subtracting. Study the order of numbers and variables below.

Phrase Translation

a number decreased by 5

x – 5

a number subtracted from 5

5 – x

13Martin-Gay, Prealgebra, 5ed