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Sect. 1.3Solving Equations Equivalent Equations Addition & Multiplication Principles Combining Like Terms Types of Equations But first: Awards, HW Review, and Play ?

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But, Before we go on…

Let’s play Name That Law!a) x + 5 + y = x + y + 5

Commutative Addition … COM +

b) 3a + 6 = 3(a + 2) Distributive … DIST

c) 7x(1 / x) = 7 Reciprocals Multiplication … RECIP x

d) (x + 5) + y = x + (5 + y) Associative Addition … ASSOC +

e) (y – 2)(3x)(y + 2) = (y – 2)(y + 2)(3x) Commutative Multiplication … COM x

f) 4(a + 2b) = 8b + 4a COM, then DIST or DIST, then COM

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Checking for Equivalent EquationsA Solution is a Replacement Value that makes an equation True

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Two Keys for Solving an Equation in One Variable We need better techniques than guessing solutions

If we Add the same number to both sides of an equation, it will still have the original solution

If we Multiply both sides of an equation by the same non-0 number, it will still have the original solution

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An Example of using Horizontal Technique for Applying the Addition Principle

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Using the Vertical Techniqueto Solve the Same Equation

6.18

7.47.4

9.137.4

y

y

6.18

7.49.137.47.4

9.137.4

2

y

y

y

WaystheCompare

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Using the Vertical Techniquecan Save Steps in more complex equations

34

611116

623117

y

yy

yy

34

663467

6347

1162311117

623117

2

y

yyyy

yy

yy

yy

WaystheCompare

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An Example of Applying the Multiplication Principle

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Combining Like Terms A Term is the product of a coefficient and

variable(s). Examples: 9x -2x2y 11 -p Like Terms have identical variable parts

Combine by adding their coefficients Example: 3xy + 11xy = (3+11)xy = 14xy Example: -2p + 6p – p = (-2+6-1)p = 3p Example: 4xyz + 6xy can’t be combined

-1

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Combining a Simple Expression

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Simplify An Expression

82013

201083

]1054[23

])2(54[23

yx

yxx

yxx

yxx

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The Opposite of an Expression

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Another “Negative” Example

764

7559

)75(59

yx

yxyx

yxyx

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Connecting Concepts:Equations vs. Expressions

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Solving Using Both Principles(First the Addition Principle, then Multiplication)

x

x

xx

xx

xxx

xxx

3

412

2323

27103

271025

27)5(25

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We have been solving Linear Equations A linear equation is one that can be reduced to

ax = b (a ≠ 0 and x is any variable to the 1st power) Don’t rush to solve them in your head Work neatly, making each step result in an equivalent equation

Every linear equation will be in one of 3 categories: Conditional – it has only one value as a solution (2x = 4) Contradiction – no value will be a solution (x = x + 1) Identity – every value will make the equation true (x = x)

Types of Linear EquationsIdentity – Contradiction - Conditional

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Solve: Is it an identity, contradiction, or a conditional equation?

Identity

xx

xxx

xxx

7272

57772

5)1(772

ionContradict

tt

tt

tt

tt

tt

tt

tt

tt

8182

99

818929

8209229

16459229

)822(59229

)8248(59229

))811(48(59229

))31(48(59229 4

lConditiona

x

x

xx

xx

2

55

53

88

7583

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What’s Next? 1.4 Introduction to Problem Solving