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

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What Is Algebra?What Is Algebra?

Algebra is a system that works from Algebra is a system that works from the known to the unknown.the known to the unknown.

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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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Algebraic ExpressionsAlgebraic Expressions

Algebraic Expressions are not solved they Algebraic Expressions are not solved they are evaluated. are evaluated.

Riemann hypothesis

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.

5

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

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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.

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

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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 ...

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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

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Def: The Def: The whole numberswhole numbers

The The whole numberswhole numbers are the are the

counting numbers:counting numbers:

0,1,2,3,4,5,6,7,8,9,10,11,12,13,…0,1,2,3,4,5,6,7,8,9,10,11,12,13,…

Def: The Def: The number linenumber line

The The number linenumber line is a picture that is a picture that represents the numbers:represents the numbers:

Numbers increase from

left to right

Property: InequalitiesProperty: Inequalities

On the number line,On the number line,

• If a number lies to the right, it is greater If a number lies to the right, it is greater • If a number lies to the left, it is lessIf a number lies to the left, it is less

4 lies to the right of 2

Notation: Symbols for Notation: Symbols for InequalitiesInequalities• We use the symbol “>” to represent We use the symbol “>” to represent

“greater than”“greater than”• We use the symbol “<” to represent We use the symbol “<” to represent

“less than”“less than”• We read our mathematical sentences We read our mathematical sentences

from left to right, just like in English.from left to right, just like in English.

Mathematical symbols representing “four is greater than two” are shown below:

4 > 2

Remember the Alligator Remember the Alligator Principle!!!Principle!!!

4 > 2

Def: Def: RoundingRounding

• RoundingRounding is an important skill in our is an important skill in our increasingly complex world.increasingly complex world.

• The use of rounding allows us to better The use of rounding allows us to better understand numbers and what they understand numbers and what they represent.represent.

• Rounding also allows us to quickly Rounding also allows us to quickly understand the magnitude of complex-understand the magnitude of complex-looking numbers.looking numbers.

Examples of RoundingExamples of Rounding

If we are talking about 6,345,989,857 If we are talking about 6,345,989,857 people, it is easier to say (and people, it is easier to say (and understand) 6.3 billion people.understand) 6.3 billion people.

If we owe $14,763.94, it is easier to say If we owe $14,763.94, it is easier to say (and understand) $15,000.(and understand) $15,000.

Method: Rounding Method: Rounding A Whole NumberA Whole Number1.1. Identify the round-off place digit (ones, Identify the round-off place digit (ones,

tens, hundreds,…).tens, hundreds,…).2.2. If the digit to the right of the round-off digit If the digit to the right of the round-off digit

place is:place is:a. a. Less than 5Less than 5, do not change the round-off , do not change the round-off place digit.place digit.b. b. 5 or greater5 or greater, increase the round-off , increase the round-off place digit by 1.place digit by 1.

3. Replace all digits to right of round-off place 3. Replace all digits to right of round-off place digit with zeros.digit with zeros.

ExamplesExamples

Def: Def: VariableVariableA letter that represents a number is A letter that represents a number is

called a called a variablevariable..

Ex:Ex:

A number plus 7 equals 21. What is that number?A number plus 7 equals 21. What is that number?

Ans.Ans. We can represent the unknown number as “X” and write this We can represent the unknown number as “X” and write this question using mathematical symbols:question using mathematical symbols:

X + 7 = 21 , X = ???X + 7 = 21 , X = ???

We see that X = 14.We see that X = 14.

Def: Def: ExpressionExpression

An An expressionexpression is a collection of numbers, is a collection of numbers, variables, and operations.variables, and operations.

3x – 4 , 9 ÷ 4 + 12 , 6xy + 4z3x – 4 , 9 ÷ 4 + 12 , 6xy + 4z

Expressions

Property: Identity Property Property: Identity Property of Zero of ZeroAdding zero to anything doesn’t change Adding zero to anything doesn’t change

the number.the number.

This property represented symbolically:This property represented symbolically:

0 + X = X , X + 0 = X0 + X = X , X + 0 = X

Ex:Ex: 0 + 5 = 5 , 5 + 0 = 50 + 5 = 5 , 5 + 0 = 5

Property: Commutative Property Property: Commutative Property of Addition of AdditionTwo numbers can be added in either Two numbers can be added in either

order with the same result:order with the same result:

a + b = b + aa + b = b + a

Ex:Ex:

4 + 9 = 9 + 44 + 9 = 9 + 4

Both equal 13!

Def: Def: SimplifyingSimplifying

When possible, it is good to make things When possible, it is good to make things more simple:more simple:

9 + 4 can be rewritten as 139 + 4 can be rewritten as 13

We We simplifiedsimplified by combining the two by combining the two numbers, 9 and 4, into one single numbers, 9 and 4, into one single number, 13.number, 13.

ExampleExample

Simplify:Simplify: 1 + 9 + X1 + 9 + X

Ans:Ans:1 + 9 + X = 10 + X1 + 9 + X = 10 + X

Note we combined the two numbers into Note we combined the two numbers into one.one.

Answer

Property: Associative Property of Property: Associative Property of AdditionAddition

When we add three or more numbers, the When we add three or more numbers, the addition may be grouped in any way.addition may be grouped in any way.

(a + b) + c = a + (b + c)(a + b) + c = a + (b + c)

Ex:Ex:

(4 + 9) + 1 = 4 + (9 + 1)(4 + 9) + 1 = 4 + (9 + 1)

13 + 1 = 4 + 1013 + 1 = 4 + 10

14 = 1414 = 14

ExampleExample

Simplify: (14 + X) + 15Simplify: (14 + X) + 15

Ans:Ans:

(14 + X) + 15 = (X + 14) + 15 (14 + X) + 15 = (X + 14) + 15 commutativecommutative

= X + (14 + 15) = X + (14 + 15) associativeassociative

= X + 29= X + 29

Def: Def: Evaluating an ExpressionEvaluating an Expression

To To evaluateevaluate an algebraic expression, we an algebraic expression, we replace the variables in the expression replace the variables in the expression with their corresponding values and with their corresponding values and simplify.simplify.

ExampleExample

Evaluate X + 2 given that X = 5.Evaluate X + 2 given that X = 5.

Ans:Ans: Replace X with 5 and simplify. Replace X with 5 and simplify.

X + 2 = 5 + 2 = 7X + 2 = 5 + 2 = 7

Answer

ExampleExample

Evaluate Evaluate aa + + bb + 7 given that + 7 given that aa equals 9 equals 9 and and bb equals 13. equals 13.

Ans:Ans: Replace Replace aa with 9, with 9, bb with 13, and with 13, and simplify.simplify.

aa + + bb + 7 = 9 + 13 + 7 = 9 + 20 = 29 + 7 = 9 + 13 + 7 = 9 + 20 = 29

Property: Subtraction is Property: Subtraction is NotNot CommutativeCommutative

3 – 2 does not equal 2 – 33 – 2 does not equal 2 – 3

“not equal to” sign

Section 1.4Section 1.4

Multiplying Whole Number Multiplying Whole Number ExpressionsExpressions

Property: MultiplyingProperty: Multiplying

• Multiplying can be thought of as Multiplying can be thought of as repeated addition.repeated addition.

Four 8’s

Def: Def: AreaArea

• AreaArea is derived using multiplication. is derived using multiplication.• A square foot is defined as the area of a A square foot is defined as the area of a

square whose sides are 1 foot long.square whose sides are 1 foot long.

Property: Area of a RectangleProperty: Area of a Rectangle• If we think of a rectangle as being If we think of a rectangle as being

composed of these boxes, we see that composed of these boxes, we see that the area is equal tothe area is equal to

Area = (Length) x (Width)Area = (Length) x (Width)

Def: Def: FactorsFactors and and ProductsProducts

• Things that are multiplied together are Things that are multiplied together are called called factorsfactors..

• The result of the multiplication is called The result of the multiplication is called the the productproduct..

Property: MultiplicationProperty: Multiplication

• Multiplication is commutative: ab = baMultiplication is commutative: ab = ba• Multiplication is associative: (ab)c = Multiplication is associative: (ab)c =

a(bc)a(bc)• Identity property of 1: 1∙a = a∙1 = aIdentity property of 1: 1∙a = a∙1 = a• Multiplication property of 0: 0∙a = a∙0 = Multiplication property of 0: 0∙a = a∙0 =

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NumbersNumbers

MonomialMonomial Any number, all by itself, is a Any number, all by itself, is a

monomial, like 5 or 2700. A monomial monomial, like 5 or 2700. A monomial can also be a variable, like “m” or can also be a variable, like “m” or “b”. It can also be a combination of “b”. It can also be a combination of these, like 98b or 78xyz. these, like 98b or 78xyz.

It cannot have a fractional or It cannot have a fractional or negative exponent. Ex negative exponent. Ex

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32

1

510

or

NumbersNumbers

BinomialsBinomials A binomial is an equation or A binomial is an equation or

expression with two terms. 3x + 1, expression with two terms. 3x + 1, 2x2x33 - 5x, x - 5x, x44 - 4, x - 19 are examples of - 4, x - 19 are examples of binomials. As well as 3x-3 = 10 or binomials. As well as 3x-3 = 10 or

5x =95x =9

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NumbersNumbers

TrinomialsTrinomials A trinomial is a polynomial with three A trinomial is a polynomial with three

terms. Examples of trinomials are terms. Examples of trinomials are

2x2x22 + 4x - 11, 4x + 4x - 11, 4x33 - 13x + 9, - 13x + 9,

7x7x33 - 22x - 22x22 + 24x, and 5x + 24x, and 5x66 - 17x - 17x22 + 97. + 97.

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