solving one-step equations

Solving One-Step Equations

Lesson 2 – 1

Addition and Subtraction Properties of Equality

a, b, and c are real numbersIf a = b…

Property Algebra ExampleAddition Property of Equality a + c = b + c x – 3 = 2

x – 3 + 3 = 2 + 3Subtraction Property of Equality a – c = b – c x + 3 = 2

x + 3 – 3 = 2 – 3

Problem 1What is the solution x + 13 = 27?

x + 13 = 27x + 13 – 13 = 27 – 13

x + 0 = 14x = 14

Why does subtracting 13 from both sides of the original equation result in an equivalent equation?

Problem 2What is the solution of -7 = b – 3?

-7 = b – 3 -7 + 3 = b – 3 + 3

-4 = b

Got it? 2bWhat is the solution of ½ = y – (3/2)?

½ = y – (3/2)½ + (3/2) = y – (3/2) + (3/2)

4/2 = y2 = y

Multiplication and Division Properties of Equality

a, b, and c are real numbersIf a = b…

Property Algebra ExampleMultiplication Property of Equality a ∙ c = b ∙ c x/3 = 2

x/3 ∙ 3 = 2 ∙ 3Division Property of Equality a / c = b / c 5x = 20

5x/5 = 20/5

Problem 3What is the solution of 4x = 6.4?

4x = 6.44x/4 = 6.4/4

1x = 1.6x = 1.6

Got it? 3aWhat is the solution of 10 = 15x?

10 = 15x10/15 = 15x/15

2/3 = 1x = x

Problem 4What is the solution of = -9?

x/4 = -9x/4 ∙ 4 = -9 ∙ 4

1x = -36x = -36

Got it? 4bWhat is the solution of x/-9 = 8?

x/-9 = 8x/-9 ∙ -9 = 8 ∙ -9

1x = -72x = -72

Problem 5What is the solution of 4/5m = 28?

4/5m = 284/5m ∙ 5/4 = 28 ∙ 5/4

1m = 35m = 35

4/5 and 5/4 are called what?

Problem 6The length of an average toucan is about two thirds of the length of a macaw. Toucans are about 24 in. long. What is the length of an average macaw?

(Toucan) = 2/3 of (Macaw)T = 2/3 ∙ M24 = 2/3M

24 ∙ 3/2 = 2/3M ∙ 3/236 = M

The average macaw is about 36 inches.

Homework

Lesson 2-1 #10 – 50 evens

Solving Two – Step Equations

Lesson 2 – 2

Problem 1What is the solution of 2x + 3 = 15?

2x + 3 = 152x + 3 – 3 = 15 – 3

2x = 122x/2 = 12/2

x = 6

Problem 2You are making a bulletin board to advertise

community service opportunities in your town. You plan to use half a sheet of construction paper for each ad. You need 5 sheets of paper for a title banner. You have 18 sheets of paper. How many ads can you make?

½ a + 5 = 18½a + 5 – 5 = 18 – 5

½ a = 13½ a ∙ 2 = 13 ∙ 2

a = 26

You can make 26 ads.

Problem 3What is the solution of (x – 7) ÷ 3 = -12?

(x – 7) ÷ 3 = -12(x – 7) ÷ 3 ∙ 3 = -12 ∙ 3

x – 7 = -36x – 7 + 7 = -36 + 7

x = -29

Look at page 90 to see another way to write this equation.

Homework


Solving Multi-Step Equations

Lesson 2 – 3

Problem 1What is the solution of 5 = 5m – 23 + 2m?

5 = 5m – 23 + 2m5 = 5m + 2m – 23

5 = 7m – 23 5 + 23 = 7m – 23 + 23

28 = 7m28/7 = 7m/7

4 = m

Check: 5 = 5(4) – 23 + 2(4)?

Problem 2What is the solution of (s + 4) + 2s = 67?

(s + 4) + 2s = 67s + 4 + 2s = 67

3s + 4 = 673s + 4 – 4 = 67 – 4

3s = 63s = 21

Check: (21 + 4) + 2(21) = 67?

Problem 3What is the solution of -8(2x – 1) = 36?

-8(2x – 1) = 36-8(2x) – (-8)(1) = 36

-16x + 8 = 36-16x + 8 – 8 = 36 – 8

-16x = 28x = -7/4

Name one mistake that could occur when solving this equation.

Problem 4Look at page 96 for Problem 4.

Problem 5What is the solution of 3.5 – 0.02x = 1.24?

3.5 – 0.02x = 1.24Multiply each term by 100 to eliminate the decimals

3.5(100) – 0.02x(100) = 1.24(100)350 – 2x = 124

350 – 350 – 2x = 124 – 350-2x = -226

x = 113

Homework


Solving Equations with Variables on Both Sides

Lesson 2 – 4

Problem 1What is the solution of 5x + 2 = 2x + 14?

5x + 2 = 2x + 145x + 2 – 2x = 2x + 14 – 2x

3x + 2 = 143x – 2 + 2 = 14 – 2

3x = 12x = 4

Problem 2What is the solution of 1.5p = 1.25p + 8?

1.5p = 1.25p + 81.5p – 1.25p = 1.25p – 1.25p + 8

0.25p = 8p = 32

Problem 3What is the solution of 2(5x – 1) = 3(x + 11)?

2(5x – 1) = 3(x + 11)10x – 2 = 3x + 33

10x – 2 + 2 = 3x + 33 + 210x = 3x + 35

10x – 3x = 3x – 3x + 357x = 35

x = 5

Problem 4aWhat is the solution of 10x + 12 = 2(5x + 6)?

10x + 12 = 2(5x + 6)10x + 12 = 10x + 12

10x – 10x + 12 = 10x – 10x + 1212 = 12

There are infinitely many solutions.

Problem 4bWhat is the solution of 9m – 4 = -3m + 5 + 12m?

9m – 4 = -3m + 5 + 12m9m – 4 = 9m + 5

9m – 9m – 4 = 9m – 9m + 5-4 = 5-4 ≠ 5

There are no solutions.

Homework


Literal Equations and Formulas

Lesson 2 – 5

Problem 1Solve the equation 10x + 5y = 80 for y.

10x + 5y = 805y = 80 – 10x

y = 80/5 – 10x/5y = 16 – 2x

Problem 2What equation do you get when you solve

ax – bx = c for x?ax – bx = cx(a – b) = cx = c/(a – b)

x = c (a – b)

“Famous Formulas”Formula Name Formula

Perimeter of a RectangleCircumference of a CircleArea of a RectangleArea of a TriangleArea of a CircleDistance TraveledTemperature C = 5/9(F – 32)

C = 2∏rA = lw

A = ½ bhA = ∏r2

d = rt

P = 2l + 2w

Problem 3What is the radius of a circle with

circumference of 64 ft? Use 3.14 for pi.

C = 2∏r64 = 2(3.14)r

64 = 6.28rr ≈ 10.2

Problem 4Write d = rt and solve for r.

d = rtd/t = r

Homework

Lesson 2-5 #12 – 40 multiplies of four

Finding Perimeter, Area and Volume

Read through page 115 – 116.

Complete the exercises 1 – 6.

Chapter 2 Mid – Chapter Quiz

Ratios, Rates and Conversions

Lesson 2 – 6

Problem 1You are shopping for T-shirts. Which store

offers the best deal?

Store A: $25 for 2 shirts

Store B: $45 for 4 shirts

Store C: $30 for 3 shirts

Problem 2Convert 330 minutes into hours.

Convert 5 ft 3 in into inches.

Problem 3The CN Tower I Toronto, Canada, is about

1815 ft tall. About how many meters tall is the tower? 1 meter ≈ 3.28 ft.

Problem 4A student ran the 50 yard dash in 5.8

seconds. At what speed did the student run in miles per hour? Round to the nearest tenth. 1 miles = 1760 yards

Homework

Lesson 2-6 #10 – 30 evens, 48 – 52

Solving ProportionsLesson 2 – 7

Cross Multiplication

A CB D

AD = BC

Problem 1What is the solution of the proportion

7 m8 12

7(12) = 8m84 = 8m10.5 = m


4 83 x

4x = 3(8)4x = 24

x = 6


b – 8 b + 35 4

5(b + 3) = 4(b – 8)5b + 15 = 4b – 32

5b – 4b + 15 = 4b – 32 – 4bb + 15 = -32

b = -47

Problem 4Look at Problem 4 on page 126.

Homework


Proportions and Similar Figures

Lesson 2 – 8

~ S ~ I ~ M ~ I ~ L ~ A ~ R~The symbol (~) means “similar to”

A F

B C

G H

Δ ABC ~ ΔFGH

~ S ~ I ~ M ~ I ~ L ~ A ~ R~AB/FG = AC/FH = BC/GH

A F

B C

G H

Δ ABC ~ ΔFGH

Problem 1AB = 10, BC = 16, EF = 12, and DF = 18

A D

B C

E F

What is the length of DE?

Problem 2 - 4Turn to page 131 to review Problems 2 -4.

Complete Lesson Check 1 – 5.

Homework

Lesson 2-8 #8 – 22

Percents Lesson 2 – 9

The Percent ProportionAlgebra

a p b 100

ExampleWhat percent of 50 is 25?

25 p 50 100

Problem 1What percent of 56 is 42?

a p b 100

42 p 56 100

42(100) = 56p4200 = 56p

75 = p

Problem 2What percent of 40 is 2.5?

p ∙ 40 = 2.540p = 2.5p = 0.625

6.25% = percent

Problem 3A dress shirt that normally cost $38.50 is on

sale for 30% off. What is the sale price of the shirt?

a = p% ∙ b= 30% ∙ 38.50

= .3(38.50)= 11.55

$38.50 - $11.55 = $26.95

Problem 4125% of what number is 17.5?

17.5 = 125% ∙ b17.5 = 1.25b

14 = b

Simple Interest FormulaI = interest P = principle

r = annual interest rate t = time

I = PrtExample:If you invest $50 at a simple interest rate of

3.5% per year for 3 years in interest you earn is

I = PrtI = 50(0.035)(3)

I = $5.25

Problem 5You deposit $840 in a savings account that

earns a simple interest rate of 4.5% per year. You want to keep the money in the account for 4 years. How much interest will you earn?

I = PrtI = 840(0.045)(4)

I = 151.2

You will earn $151.2.

Homework

Lesson 2-9 #9 – 42 multiplies of three

Changed Expressed as a Percent

Lesson 2 – 10

Percent Change

amount of increase or decrease

% Change = original amount

Amount of increase = new – original

Amount of decrease = original – new

Problem 1A coat is on sale. The original price of the

coat is $82. The sale price is $74.50. What is the discount expressed as a percent of change?

82 – 74.50 = 7.5

7.5/82 = 0.09

9%

Problem 2A store buys an electric guitar for %295.

The store then marks up the price or the guitar to $340. What is the markup expressed as a percent change?

340 – 295 = 45

45/340 = 0.15

15%

Relative Error or Percent Error

│estimated value – actual value│=

actual value

Problem 3A decorator estimates that a rectangular

rug is 5ft by 8ft. The rug is actually 4f by 8 ft. What is the percent error in the estimated area?

│estimated value – actual value││5(8) – 4(8)│ = 8

8/32 = 0.25

25% error

Problem 4You are framing a poster and measure the

length of the poster as 18.5 in to the nearest half inch. What are the minimum and maximum possible lengths of the poster?

.5/2 = .2518.5 - .25 = 18.25

18.5 + .25 = 18.75

The min is 18.25 in, the max is 18.75 in

Problem 5Turn to page 147 to view Problem 5.

Homework


Chapter 2 Pull It All Together – 50 points

Choose 2 Tasks and create a cross word puzzle from 20 vocabulary words

solving one-step equations

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