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

ALGEBRA I(A) CH3 NOTESWinter, 2010-2011

Ms. Ellmer

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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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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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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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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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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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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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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.

CH3-1 SOLVING TWO-STEP EQUATIONS

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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

CH3-1 SOLVING TWO-STEP EQUATIONS

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ALWAYS WORK DOWN YOUR PAGE

CH3-1 SOLVING TWO-STEP EQUATIONS

Now you try

EVENS 2-34

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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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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.

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.

CH3-2 SOLVING MULTI-STEP EQUATIONS

Now, you try

ODDS 1- 49

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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.

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.

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.

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.

CH 3-3 EQUATIONS W/VARIABLES ON BOTH SIDES

Now, you try

EVENS 2-38

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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

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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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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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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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

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.

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

CH 3-4 RATIO AND PROPORTION

Now, you do

ODDS 1- 45

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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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Ex.

8 corresponds to ______ 1210 corresponds to _______ 22

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3-5 PROPORTIONS AND SIMILAR SHAPES

8

10

12

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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

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2.5x

3

Now, you do

1-8 ALL PROBLEMS!!!!!

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3-5 PROPORTIONS AND SIMILAR SHAPES

CH 3-6 EQUATIONS AND PROBLEM SOLVING

You do story problems

1-13 ODDS

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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

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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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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CH 3-7 PERCENT OF CHANGE

Now you do

EVENS 2-26

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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

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

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CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Now you do

ODDS

1-19 in 10 minutes!

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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

CH 3-8 FINDING AND ESTIMATING SQUARE ROOTS

Now you do

ODDS

21,27,31-43

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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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Let’s label the sides of this triangle:

CH 3-9 THE PYTHAGOREAN THEOREM

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Pythagorean Theorem:

c2 = a2 + b2

NOTE: Must be a Right Triangle!

CH 3-9 THE PYTHAGOREAN THEOREM

ca

b

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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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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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NOW, YOU DO

EVENS

2-30

CH 3-9 THE PYTHAGOREAN THEOREM

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

CH 3

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