a lgebra i(a) ch3 n otes winter, 2010-2011 ms. ellmer 1

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ALGEBRA I(A) CH3 NOTES Winter, 2010-2011 Ms. Ellmer 1

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Page 1: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

ALGEBRA I(A) CH3 NOTESWinter, 2010-2011

Ms. Ellmer

1

Page 2: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONSBackground:

By writing an algebraic equation, you can use it for any values of the variable. So it has endless uses!

Vocab:Equation: Expression with an = sign“Solve for x:” x is by itself on one side of = signTerm: a number, a variable, or the product of a number and any variable(s).“Combine Like Terms:” x’s combine with x’s, y’s combine with y’s, x2 combines with x2, numbers combine with numbers ONLYOpposite Functions: called inverse operations, operations that undo each other.

Opposite Functions:+ -* ∙ /x2 √x2

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Page 3: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

IN ORDER TO SOLVE EQUATIONS, YOU MUST BE ABLE TO COMBINE LIKE TERMS PROPERLY!!!!

LET’S PRACTICE!

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Page 4: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

Ex.1 Combine all like terms in the following expressions.

a. 2x + 362x + 36

b. (42+4x) + 83 4x +125

c. (42+4x) + 83 + 3(y-3)4x +3y +116

d. 4x+3y+2x2+16+9y2x2 +4x +12y +16

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Page 5: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

IN ORDER TO SOLVE EQUATIONS, YOU MUST BE ABLE TO IDENTIFY AND USE OPPOSITE

FUNCTIONS CORRECTLY!!!!

LET’S PRACTICE!

You must use the OPPOSITE FUNCTION to move stuff from one side of the = to the other.

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Page 6: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

Ex.2: Identify the opposite function needed to get the variable isolated to one side.

a. x – 6 = 13 +6 = +6ADDING

b. 2x = 52---- = ----

2 2DIVIDING

c. x + 8 = 56 - 8 = -8SUBTRACTING

d. x2 = 36√ x2 = √36TAKE THE SQUARE ROOT

Opposite Functions+ -*∙ /x2 √x2

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Page 7: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

Now that you know what “Combine Like Terms” means and what opposite functions do, you

can follow a simple recipe to solve equations.

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Page 8: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

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Recipe to Solve EquationsStep1: Get x term(s) alone on one side.Step2: Combine Like Terms.Step3: Isolate x using opposite functions.Step4: Plug x value back in to original

question and check answer.

Page 9: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

9

Ex.3 Solve the equation.

36.9 = 3.7b – 14.9

+14.9 = +14.9 Step 1 & 2 51.8 = 3.7b ----- = ----- Step 3 3.7 3.7 14 = b

36.9 = 3.7(14) – 14.9 Step 4 36.9 = 36.9

Page 10: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

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ALWAYS WORK DOWN YOUR PAGE

Page 11: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-1 SOLVING TWO-STEP EQUATIONS

Now you try

EVENS 2-34

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Page 12: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-2 SOLVING MULTI-STEP EQUATIONS

If you follow the recipe, it doesn’t matter how big the equation gets….you can always know what to do.

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Page 13: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-2 SOLVING MULTI-STEP EQUATIONS

Ex. 1 Solve each equation. Check your answer.

2w - 5w + 6.3 = -14.4 -6.3 = - 6.3

2w – 5w = -20.7-3w = -20.7----- -------3 -3 w = 6.9

2(6.9) -5(6.9) + 6.3 = -14.413.8-34.5+6.3 = -14.4-14.4 = -14.4

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Recipe to Solve EquationsStep1: Get x term(s) alone on one

side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 14: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-2 SOLVING MULTI-STEP EQUATIONS

Ex.2 Solve each equation. Check your answer.

2(m+1) = 162m +2 = 16 -2 = -22m = 14---- ----2 2m = 7

2(7+1) = 162(8) = 1616 = 16

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Recipe to Solve EquationsStep1: Get x term(s) alone on one

side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 15: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH3-2 SOLVING MULTI-STEP EQUATIONS

Now, you try

ODDS 1- 49

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Page 16: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

What do we do if we have variables on both sides of the = sign??????

THE SAME RECIPE!

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Recipe to Solve EquationsStep1: Get x term(s) alone on one

side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 17: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

Ex. 1 Solve each equation. Check your answer. Write identity and no solution if applicable.

#1. 7 – 2n = n – 14 -n = -n 7 – 3n = -14 -7 = - 7 - 3n = -21 ----- = ------ -3 -3 n = 7Plug it back in to check answer: -7 = -7

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Recipe to Solve EquationsStep1: Get x term(s) alone on

one side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 18: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

Ex. 1 Solve each equation. Check your answer. Write identity and no solution if applicable.

#7. 3(n-1) = 5n +3 -2n 3n -3 = 3n + 3 -3n = -3n -3 = +3Where did n go?Canceled out, soNO SOLUTION

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Recipe to Solve EquationsStep1: Get x term(s) alone on

one side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 19: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

Ex. 1 Solve each equation. Check your answer. Write identity and no solution if applicable.

#5. 8z – 7 = 3z – 7 + 5z 8z – 7 = 8z – 7Is this the same exact thingOn the left side as right Side? If yes, then

IDENTITY

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Recipe to Solve EquationsStep1: Get x term(s) alone on

one side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 20: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

Now, you try

EVENS 2-38

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Page 21: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTIONBackground:

Ratios and proportions have many uses in many industries. They can be used to read a map, mix chemicals in painting and landscaping, mix cleaners in home improvement projects, scaled drawings, and finding unit prices while grocery shopping.

Vocabulary:Ratio: A comparison of two numbers. Written in 3 ways:

1. a to b

2. a:b

3. a

b

Unit Rate: Any number over 1 with units “something per something else” 21

Page 22: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTIONHow to Use It:

In Science, unit rates allow you to “cancel your units,” or use dimensional analysis to get the units you want.

Ex.1

40.56(km) ∙ (1 mi) = (hr) 1.6 (km)

25.35 mi/hr …..on a 10 speed bike!!!!!

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Page 23: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTIONEx.2 Page 143 Lance Armstrong!

In 2004, Lance Armstrong won the Tour de France completing the 3391 km course in about 83.6 hours. Find Lance’s average speed using v=d/t.

d=3391 kmt = 83.6 hrv = ?v = d

tv = (3391 km)

(83.6 hr)v = 40.6 km/hr

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Page 24: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTIONVocabulary Continued:

Proportion: is an equation that states that two ratios are equal, written as:

a = c b d

And you read it as, “a is to b as c is to d”

What is the difference between a set of ratios and a proportion?????

THE = SIGN IS IN THE PROPORTION ONLY!!!!!!

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Page 25: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTION

Ex.3 Solve for x.

1:16 = ? : 36 1 = x

16 36 What should we do now?

Yep, cross multiply and start flexing your algebra muscles!

x = 2.2525

Page 26: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTIONThe proportions can get really big and have variables….no

problemo!Ex. 4 Solve each proportion.2X-2 = 2X-4 14 6

6(2X-2) = 14(2X-4) 12x – 12 = 28x – 56-28x -28x-16x – 12 = -56 + 12 = +12-16x = -44 x = 2.75

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Recipe to Solve EquationsStep1: Get x term(s) alone on

one side.Step2: Combine Like Terms.Step3: Isolate x using opposite

functions.Step4: Plug x value back in to

original question and check answer.

Page 27: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTION

Are we done?

Nope, go back in and check your answer….

2(2.75)-2 = 2(2.75)-4 14 6

0.25 = 0.25

YES!!!!!27

Page 28: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-4 RATIO AND PROPORTION

Now, you do

ODDS 1- 45

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Page 29: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

3-5 PROPORTIONS AND SIMILAR SHAPES

Background: Proportions can help determine the dimensions for certain objects, by comparing two similar shapes. This method is used by architects, designers, and computer aided drafting (CAD) software.

Vocabulary: Corresponding side: a side of one object that can be compared to a side of another object in the same location/dimension.

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Page 30: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Ex.

8 corresponds to ______ 1210 corresponds to _______ 22

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3-5 PROPORTIONS AND SIMILAR SHAPES

8

10

12

22

Page 31: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

How to Use It:Ex.1 Each pair of figures is similar. Find the length of x.

2.5 = 5 x 32.5(3) = 5(x)7.5 = 5x 5 51.5 = x

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3-5 PROPORTIONS AND SIMILAR SHAPES

5

2.5x

3

Page 32: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Now, you do

1-8 ALL PROBLEMS!!!!!

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3-5 PROPORTIONS AND SIMILAR SHAPES

Page 33: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-6 EQUATIONS AND PROBLEM SOLVING

You do story problems

1-13 ODDS

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Page 34: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-7 PERCENT OF CHANGE

Background: Describing relationships using percents can be seen in shopping/pricing, engine efficiencies and grades.

Vocabulary:Percent of change: the ratio of the amount of change over the original amount, or % change = amount of change * 100%

original amount% error = estimated – actual * 100%

actual 34

Page 35: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-7 PERCENT OF CHANGEHow To Use It:

Ex.1 Find the percent of change. Describe it as an increase or decrease.

40 cm to 100 cmAmount of change = 100 cm – 40 cm = 60 cm

% change = amount of change * 100% original amount

% change = 60 cm * 100% 40cm

% change = 1.5 * 100%%change = 150% INCREASE

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Page 36: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-7 PERCENT OF CHANGEHow To Use It:

Ex.2 A student estimated the mass of the Physics textbook to be 1750 grams. After measuring the mass on a triple beam balance, the actual textbook mass was 2450 grams. Find the student’s percent error.

% error = estimated – actual * 100% actual

% error = 1750 – 2450 * 100% 2450

%error = 28.6%

Do you want a high % error or a low % error?

LOW! < 10% in industry is EXCELLENT!

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Page 37: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-7 PERCENT OF CHANGE

Now you do

EVENS 2-26

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Page 38: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Background: Square roots and perfect squares are used in many problems including construction and science lab work.

Vocabulary:Square root: a special case for a number, that when multiplied by itself, or squared, it gives that number.Ex. 42 = 16, so 4 and -4 are square roots of 16Symbol = √ with a number under this symbolYOU CAN’T HAVE A – SIGN UNDER √Perfect Square: squares of integers

38Integer

1 2 3 4 5 6 7 8 9 10

Square

1 4 9 16 25 36 49 64 81 100

Page 39: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

How To Use It:Ex.1 Use a calculator to find the value of each square root.

a. +/- √38

+6.2 or -6.2

b. √19.38

4.4

c. √400

20

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CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Page 40: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Now you do

ODDS

1-19 in 10 minutes!

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Page 41: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

You can also simplify a radical instead of solving it.To simplify, look for perfect squares to be taken out of

square root sign.

How To Use It:Ex.1 Simplify the expression.

√32

√8∙4

√2∙4 ∙4

2∙2√2

4√241

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Page 42: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Now you do

ODDS

21,27,31-43

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Page 43: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Background:This dude, Pythagoreas lived around 500 B.C. He developed a way to determine the lengths of sides of a right triangle & applied it in the construction industry.

Vocabulary:Right triangle: A with one 90° angleHypotenuse: The longest leg of a Right and IS ALWAYS ACROSS FROM THE RIGHT ANGLE BOX

Symbol: c Legs: The other two sides of a Right Symbols: a and b

CH 3-9 THE PYTHAGOREAN THEOREM

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Page 44: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Let’s label the sides of this triangle:

CH 3-9 THE PYTHAGOREAN THEOREM

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Page 45: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Pythagorean Theorem:

c2 = a2 + b2

NOTE: Must be a Right Triangle!

CH 3-9 THE PYTHAGOREAN THEOREM

ca

b

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Page 46: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

How to Use It:

Ex.1 Find the missing side of the right triangle. a=12 b=? c=35c2=a2+b2

352=122+b2

1225=144+b2

1081=b2

√1081=√b2

32.9 = b

CH 3-9 THE PYTHAGOREAN THEOREM

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Page 47: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

How to Use It:

Ex.2 Determine whether the given lengths are sides of a right triangle.

20,21,29c2=a2+b2

292=202+212

841=400+441841 = 841YES

CH 3-9 THE PYTHAGOREAN THEOREM

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Page 48: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

NOW, YOU DO

EVENS

2-30

CH 3-9 THE PYTHAGOREAN THEOREM

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Page 49: A LGEBRA I(A) CH3 N OTES Winter, 2010-2011 Ms. Ellmer 1

Yeah! We are done with CHAPTER 3!!!!!!!

CH 3

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