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
Lesson 2-2 #12 – 40 evens
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
Lesson 2-3 #10 – 52 evens
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
Lesson 2-4 #10 – 32 evens
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
Problem 2What is the solution of the proportion
4 83 x
4x = 3(8)4x = 24
x = 6
Problem 3What is the solution of the proportion
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
Lesson 2-7 #10 – 36 evens
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
Lesson 2-10 #8 – 32 evens
Chapter 2 Pull It All Together – 50 points
Choose 2 Tasks and create a cross word puzzle from 20 vocabulary words