Algebra 1A UNIT 2: EQUATIONS
LESSON 2-1: Solving One-Step Equations
LESSON 2-2: Solving Two-Step Equations
LESSON 2-3: Solving Multi-Step Equations
LESSON 2-4: Solving Equations with Variables on Both Sides
LESSON 2-5: Literal Equations and Formulas
LESSON 2-6: Ratio, Rates, and Conversions
LESSON 2-7: Solving Proportions
LESSON 2-8: Proportions and Similar Figures
LESSON 2-9: Percents
LESSON 2-10: Change Expressed as a Percent
LESSON 2-1: Solving One-Step Equations

OBJECTIVE: To solve one-step equations in one variable.
_________________ ______________ are equations that have the SAME solution(s).
You can find the SOLUTION of a one-step equation using the PROPERTIES of EQUALITY and
INVERSE OPERATIONS to write a similar equivalent equation.
An inverse operation undoes another operation.
Examples:
PROPERTIES OF EQUALITY:
Addition Property of Equality: Adding the same number to each side of an equation produces equivalent equation.
Algebra: Example:
Subtraction Property of Equality: Subtracting the same number from each side of an equation produces an equivalent equation.
Algebra: Example:
Multiplication Property of Equality: Multiplying each side of an equation by the same nonzero number produces an equivalent equation.
Algebra: Example:
Division Property of Equality: Dividing each side of an equation by the same nonzero number produces an equivalent equation.
Algebra: Example:
To SOLVE an equation, you must ISOLATE the variable. You do this by getting the variable with a
COEFFICIENT of 1 alone on one side of the equation.
To CHECK your solution, SUBSTITUTE your answer back into the equation and see if it makes a
TRUE statement.
Examples 1a
– 1n:
Solve and check each equation. a.) -7 = b – 3 b.) m – 8 = -14
1 c.)
2
 y

3
2

d.) 15 + s = 42 f.) 4x = 6.4 i.) 2

3 x
5 d.) -12 + x = 17 g.) 10 = 15x j.)

9
 x
4 e.) x + 13 = 27 h.) -3.2z = 14 k.) r
3

19 l.)

12
 x
5 m.) 3

1 x
3 n.) x

9

8
When the coefficient of the variable in an equation is a FRACTION, you can use the RECIPROCAL of the fraction to solve the equation.
Examples 2a – 2c: Solve each equation.
5 a.)
2 x

15 b.)
4
5 m

28 c.) 12

3
4 x

Example 3: An online DVD rental company offers gift certificates that you can use to purchase rental plans. You have a gift certificate for $30. The plan you select costs $5 per month. How many months can you purchase with the gift certificate?
Example 4: Rob is driving at 40 miles per hour. Stacey is driving 3/2 times as fast. How fast is
Stacey driving?
Example 5 : A hardcover book costs $19 more than its paperback edition. The hardcover book costs
$26.95. How much does the paperback cost?
Example 6 : Annie and Michael tried to solve the equation x + 4 = -9. Who solve the equation correctly? Explain.
Annie: x + 4 = - 9
+ 4 + 4
x = - 5
Michael: x + 4 = -9
-4 -4
x = -13
Example 7: Find the mistake. Then solve the equation and find the correct answer.
 x
9
 
3

9
 x

9
 
3
 
9 x
 
27

LESSON 2-2: Solving Two-Step Equations
OBJECTIVE: To solve two-step equations in one variable.
BELL RINGER: Solve each equation. a.) -35 = -7b b.) n
5
 
13 c.) d + 9 = -12 d.) f
– 8 = 15
NOTES
Example 1: Suppose you are ordering roses online. Roses cost $5 each, and shipping costs $10.
Your total cost depends on how many roses you buy. What two operations are involved?
Steps to Solve Two-Step Equations: Undo operation in the reverse order of the order of operations.
1.) Undo Addition and Subtraction
2.) Undo Multiplication and Division
Examples 2a – 2i: Solve each equation. a.) 2x + 3 = 15 b.) 5
 t
2

3 c.) 2x + 11 = 51 d.) 5n – 18 = -33 e.) n
3

(

11 )

16 f.) 6r + 19 = 43 g.) a
5

4

10 h.) -8y – 28 = -36 i.) x
5

35

75

Example 3 : You 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 construction paper for a title banner. You have 18 sheets of construction paper. How many ads can you make?
Example 4: A train is 100 miles away from a station. It is approaching the station at 40 miles per hour. How long will it take for the train to arrive?
Example 5: Suppose you buy a jumbo lemonade for $1.50 and divide the cost of an order of chicken wings with two friends. Your share of the total bill is $5.50. Write and solve an equation to find the cost of the chicken wings?
Example 6: Solve each word problem.

When one side of an equation is a fraction with more than one term in the numerator, you can still undo division by multiplying each side by the DENOMINATOR.
Examples 8a – 8c: Solve each equation. a.) x

3
7
 
12 b.) 6
 x

2
4 c.) x

5
6

15

When you use DEDUCTIVE REASONING, you must state your step and your reason for each step using properties, definitions, or rules.
Example 9: What is the solution of –t + 8 = 3? Justify each step.
Example 10: What is the solution of 10 + 2x = 20? Justify each step.
Example 11: What is the solution of x
3

5

4 ? Justify each step.
Example 12 : Find the mistake. Then solve the equation to find the correct answer.
2x – 6 = 8
-6 -6
2x = 2
x = 1

LESSON 2-3: Solving Multi-Step Equations
OBJECTIVE: To solve multi-step equations in one variable.
BELL RINGER: Solve each equation. a.) 3t – 4 = 44 b.) x
5

6
 
1
NOTES
1.) Simplify each side of the equation (distribute, combine like terms, etc.)
2.) Move the constants to the other side of the equation and combine like terms.
3.) Solve the resulting equation.
Examples 1a
– 1f:
Solve each equation. a.) 5 = 5m – 23 + 2m b.) 11m – 8 – 6m = 22 c.) -2y + 5 + 5y = 14 d.) -8(2x
– 1) = 36 e.) 18 = 3(2x
– 6) f.) 12(3
– x) = 60
***To solve equations containing fractions, clear fractions by multiplying both sides of the equation by the common denominator . Essentially, this means that EVERY TERM on both sides will be multiplied by the common denominator.

Examples 2a
– 2c:
Solve each equation. a.)
3 x
4
 x
3

10 b.)
2 b
5

3 b
4

3
1 c.)
9

5
6
 m
3
***To solve equations containing decimals, clear decimals by multiplying both sides of the equation by a power of 10 that makes all coefficients integers.
Examples 3a – 3c: Solve each equation. a.) 3.5 – 0.02x = 1.24 b.) 0.5x – 2.325 = 3.95 c.) 2.3 – 0.05x = 4.80
Example 4: Noah and Kate are shopping for new guitar strings in a music store. Noah buys 2 packs of strings. Kate buys 2 packs of strings and a music book. The music book costs $16. Their total cost is $72. How much is one pack of strings.
Example 5: Jason bought 5 movie tickets and a bag of popcorn. Jeremy bought 2 movie tickets.
They spent $54 total. If the popcorn costs $5, how much did each movie ticket cost?
LESSON 2-4: Solving Equations with Variables on Both Sides

OBJECTIVE: To solve equations with variables on both sides.
To identify equations that are identities or have no solutions.
BELL RINGER: Solve each equation. a.) -2(j – 12) = 18 b.) 7x + 14 – 3x = 30
NOTES
1.) Simplify each side of the equation (distribute, combine like terms, etc.)
2.) Move the variables to one side of the equation and combine like terms.
3.) Move the constants to the other side of the equation and combine like terms.
4.) Solve the resulting equation.
Examples 1a
– 1j:
Solve each equation. a.) 5x + 2 = 2x + 14 b.) 7k + 2 = 4k – 10 c.) 5s + 13 = 2s + 22 d.) 2(5x
– 1) = 3(x + 11) e.) 4(2y + 1) = 2(y
– 13) g.) 2(3x + 1) = 4(x
– 5) h.) 10m + 8 = 7m + 17
Inequalities can have 3 types of solutions:
*ONE solution Example: f.) 7(4
– a) = 3(a – 4) i.) 3(4a + 2) = 2(a
– 2)

*NO solution
*INFINITE number of solutions
(ALL REAL NUMBERS)
Example:
Example:
Examples 2a – 2f: Solve each equation. b.) 10x + 12 = 2(5x + 6) a.) t + 8 = -t + 6 + 2t d.) 3(4b
– 2) = -6 + 12b e.) 2x + 7 = - (3
– 2x) c.) 9m
– 4 = -3m + 5 + 12m f.) 3x + 7 = -3x + 2 + 6x
Example 3: An office manager spent $650 on a new energy-saving copier that will reduce the monthly electric bill for the office from $112 to 88. In how many months will the copier pay for itself?
Example 4: It takes a graphic designer 1.5 h to make one page of a Web site. Using new software, the designer could complete each page in 1.25 h, but it takes 8 h to learn the software. How many
Web pages would the designer have to order to save time using the new software?
Example 5 : Lorelle pays someone $35 to clear her driveway when it snows. She wants to buy an electric snow blower for $689 so that she can clear her own driveway. If the area Lorelle lives in has an average of 5.2 snowfalls each season, how many seasons will it take her to break even on what she spent for the snow blower?

LESSON 2-5: Literal Equations and Formulas
OBJECTIVE: To rewrite and use literal equations and formulas.
BELL RINGER: Solve each equation. a.) 5(x + 4) = x + 2x + 6 b.) 2a + 3 = ½(6 + 4a)
NOTES
A LITERAL EQUATION is an equation that involves two or more variables.
Example 1: The equation 10x + 5y = 80, where x is the number of pizzas and y is the number of sandwiches, and a budget of $80. How many sandwiches can you buy if you buy 3 pizzas? 6 pizzas?
Example 2 : Solve the equation 3x + y = 10 for y. What does y equal if x = 1? If x = 3?
Example 3: 4 = 2m – 5n for m. What are the values of m when n = -2, 0, and 2?

Example 4 : What equation do you get when you solve ax – bx = c for x?
Example 5 : What equation do you get when you solve –t = r + px for x?
Example 6: What equation do you get when you solve a x
 b x
 c for x?
A FORMULA is an equation that states a relationship among quantities. Formulas are special types of
LITERAL EQUATIONS.
Notice that some formulas use the same variables, but the definitions of the variables are different
Formula Name
Perimeter of a Rectangle
Formula
P = 2l + 2w
Definition of Variables
P = perimeter, l = length, w = width

Circumference of a Circle
Area of a Triangle
Area of a Rectangle
Area of a Circle
Distance Traveled
Temperature
C = 2
πr
A = ½bh
A = lw
A = πr 2 d = rt
C = circumference, r = radius
A = area, b = base, h = height
A = area, l = length, w = width
A = area, r = radius d = distance, r = rate, t = time
C = degrees Celsius,
F = degree Fahrenheit
C

5
9

F

32

Example 7: What is the radius of a circle with circumference 64 ft? Round to the nearest tenth.
Example 8 : What is the height of a triangle that has an area of 24 in.
2 and a base with a length of 8 in.?
Example 9 : What is the length of a rectangle with area 98 square inches and base 14 inches?
Example 10 : Karen walks 3 miles per hour and covers 12 miles. For how long did she walk?
Example 11: The monarch butterfly migrates 1700 miles annually. It takes a typical butterfly about
120 days to travel one way. What is the average rate at which a butterfly travels in miles per day?
Round to the nearest mile per day.

Example 12: The whales travel a distance of about 5000 mi each way at an average rate of 91 miles per day. About how many days does it take the whales to migrate one way?
Example 13:

Example 14:
LESSON 2-6: Ratios, Rates, and Conversions
OBJECTIVE: To find ratios and rates.
To convert units and rates.
BELL RINGER: A triangle has a base of 7 cm and an area of 28 cm 2 . What is its height?
NOTES
A ________________ is a comparison of two quantities. You can write a ratio in three ways.
Example: Algebra:
Two Ratios that name the same number are _____________ RATIOS. You can find ____________
RATIOS by writing a ratio as a ______________ and finding an ___________________ FRACTION.
A ______ is a ratio that compares two quantities measured in ____________ units.

Example:
The rate for _______ unit of a given quantity is the_______ _______. To find the unit rate, _______ the first quantity by the second quantity.
Example:
A ________________ ___________ is a ratio of two equivalent measures in different units. A conversion factor is always equal to 1.
The process of converting units is called ________ ____________ or _______________
__________.
Example 1: You 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
Example 2: If Store B lowers its price to $42 for 4 shirts, does the solution to the previous example change? Explain.
Example 3: You are shopping for sweaters. Which store offers the best deal?
Store A
$60 for 2 sweaters
Store B
$80 for 3 sweaters
Examples 4a
– 4f:
What is the given amount converted to the given units. a.) 330 min; hours b.) 15 kg; grams c.) 5 ft 3 in.; inches d.) 1250 cm; meters e.) 5 lbs 8 oz; lbs f.) 40 ounces; lbs

Example 5: The CN Tower in Toronto, Canada, is about 1815 ft tall. About how many meters tall is the tower? Use the fact that 1 m = 3.28 ft.
Example 6 : The Space Needle in Seattle, Washington, is 605 ft tall. About how many meters tall is the building? Use the fact that 1 m = 3.28 ft.
Example 7 : A student ran 100 yd in 12 s. At what speed did the student run in miles per hour?
Example 8: An athlete ran a sprint of 100 ft in 3.1 s. At what speed was the athlete running in miles per hour? Round to the nearest mile per hour?
Example 9 : A student ran the 50-yd dash in 5.8 s. At what speed did the student run in miles per hour? Round your answer to the nearest tenths.
LESSON 2-7: Solving Proportions
OBJECTIVE: To solve and apply proportions.
BELL RINGER: If a football field is 100 yd long, what is the length in miles?
NOTES

A ______________ is an _____________ stating that two ratios are _______.
Example: Algebra:
Examples 1a – 1c: Use the Multiplication Property of Equality to find the solution to each proportion. a.)
7
8
 m
12 x b.)
7

4
5 c.)
5
6
 n
11
CROSS-PRODUCT PROPERTY:
If two ratios from a proportion, the cross products are equal. If two ratios have equal cross products, they form a proportion.
Example: Algebra:
Examples 2a – 2c: Use cross products to find the solution to each proportion. y a.)
3

3
5
4 b.)
3

8 x
3 c.)
5

13 b
Examples 3a – 3c: What is the solution to the following proportions? a.) b

5
8
 b

4
3 b.) n
5

2 n
6

4 c.) x

2
3
 x

3
5
Example 4: Annie used the cross product property to determine if ratio form a proportion. Is her work correct? Explain.

Example 8:
Example 9:
3
4
3

12

12
16

36

64
4

16
Example 5: Ski socks are on sale at 3 for $40. What is the price of one pair?
Example 6 : An 8-oz can of orange juice contains about 97 mg of vitamin C. About how many milligrams of vitamin C are there in a 12-oz can of orange juice?
Example 7 : A portable media player has 2 GB of storage and can hold about 500 songs. A similar but larger media player has 80 GB of storage. About how many songs can the larger media player hold

Example 10:
LESSON 2-8: Proportions and Similar Figures

OBJECTIVE: To find missing lengths in similar figures.
To use similar figures when measuring indirectly.
BELL RINGER: Solve each proportion.
72 a.)
45

8 m b b.)
18

9
5
NOTES
When two figure have the SAME ________, but ________ necessarily the SAME _____, they could be SIMILAR. SYMBOL:
In SIMILAR TRIANGLES, corresponding _________ have the _______ measure. The corresponding
_________ are ________________.
Examples 1a – 1b: Name the corresponding angles and sides of the similar triangles.

a.) ΔTDJ and ΔRFC b.) ΔRPQ and ΔUST
R Q
S
U
T
P
Examples 2a – 2d: Find the length of the missing sides in the similar triangles. a.) ΔABC and ΔYXZ are similar triangles.
n m b.) ΔTSV and ΔQPR are similar triangles.
c.) The two trapezoids are similar.
z x y d.) The two quadrilaterals are similar.

y
z
You can use __________ __________________ to measure distances that are difficult to measure directly. You do this by using proportions and similar triangles.
Example 3: A 6-ft-tall person standing near a flagpole casts a shadow 4.5 ft long. The flagpole casts a shadow 15 ft long. What is the height of the flagpole?
Example 4: A 4 ft high wall casts a 6 ft shadow. A tree near the wall casts a 24 ft shadow. What is the height of the tree?
A _______ _____________ is an enlarged or reduced drawing of an object that is similar to actual object.
A _________ is the ratio that compares a length in a drawing or model to the corresponding length in the actual object.
Example 5: You have a scale drawing of a boat. The length of the boat on the drawing is 3 cm.
What is the actual length of the boat?

Example 6: The scale of a drawing is 1 in : 6 ft. Find the actual length for a drawing length of 4.5 in.
Example 7: Find the actual distance from Charlotte to Winston-Salem.
Example 8: Estimate the actual distance from Charlotte to Raleigh.
You can use the ______ to find the scale of a drawing or a model.
Example 9: Refer to the model boxcar shown below. The actual length of a boxcar is 609 in. What is the scale of the model?
Example 10: The actual length of the wheelbase of a mountain bike is 260 cm. The length of the wheelbase in a scale drawing is 4 cm. Find the scale of the drawing?

Example 11: You want to make a scale model of a sailboat that is 51 ft long and 48 ft tall. You plan to make the model 17 in. long. What is the height of the model?
Example 12: An architect’s model of a house is 44 in. high. The actual house is 20 ft high and 45 ft wide. What should the width of the architect’s model be?
Example 13:
The key for a map is ¼ in. represents an actual distance of 5/8 mi. Find the actual distance represented by 1 in. to write the scale of the map.
Example 14: The scale on a map shows that a map distance of 2/5 in. represents an actual distance of 4/5 mi. Find the actual distance represented by 1 in. to write the scale of the map.
LESSON 2-9: Percents
OBJECTIVE: To solve percent problems using proportions.
To solve percent problems using percent equations.

BELL RINGER: A scale drawing of a playground is 4.7 cm wide and 6.2 cm long. The playground is actually 15 m wide. How long is the playground?
NOTES
Ratio is Percent is
Example 1 : What percent of 2,000 is 204? Example 2 : What percent of 92 is 23?
Example 3 : What percent of 150 is 45?
Example 5 : 35% of 90 is what number?
Example 4
Example 6
: 20% of 55 is what number?
: 85% of 20 is what number?
Example 7: 117 is 45% of what number?
Example 8: Suppose your entertainment budget is 30% of your weekly wages from a job. You plan to spend $10.50 on a movie night. How much will you need to earn at your job in order to stay within your budget?

Example 9: Your math teacher assigns 25 problems for homework. You have done 60% of them.
How many problems have you done?
You can translate percent problems into equations to find parts, wholes, or percents.
OF mean
IS means
Example 10 : A ski resort in New Hampshire begins the season with 60% of its trails open. There are
27 trails open. How many trails does the ski resort have?
Example 11: A dress shirt that normally costs $38.50 is on sale for 30% off. What is the sale price of the shirt?
Example 12: A plane flies with 54% of its seats empty. If 81 seats are empty, what is the total number of seats on the plane?
Examples 13a – 13f : Use an equation to solve the following. a.) 18% of what number is 81? b.) What number is 39% of 377?

c.) 32% of 40 is what number? e.) What percent of 200 is 45? d.) 27% of 60 is what number?
Example 14: Of 3,072 teens surveyed, 2,212 say they read for fun. What percent of the teens surveyed say they read for fun?
Example 15 : It rained 75 days last year. About what percent of the year was rainy?
When you first deposit money in a savings account, you deposit is called ______________.
The bank takes the money and invests it. In return, the bank pays you ___________ based on the
___________ _______.
SIMPLE INTEREST is interest paid on the ____________.
Formula:
I = p = r = t =
Examples 16a – 16b: Suppose you deposit $500 in a savings account. The interest rate is 4% per year. a.) Find the simple interest earned in 6 years. Find the total of the principle plus interest. b.) Find the simple interest earned in 3 months. Find the total of the principle plus interest.

Examples 17a – 17b : Suppose you deposit $1,000 in a savings account. The interest rate is 6% per year. a.) Find the simple interest earned in 2 years. Find the total of the principle plus interest. b.) Find the simple interest earned in 6 months. Find the total of the principle plus interest.
Example 18: Find the simple interest you pay on a $220 loan at a 5% annual interest rate for 4 years.
Example 19: You invest $2,000 in simple interest account. The balance after 8 years is $2,720.
What is the interest rate?
Example 20: You deposited $125 in a savings account that earns a simple interest rate of 1.75% per year. You earned a total of $8.75 in interest. For how long was your money in the account?
Example 21:

LESSON 2-10: Change Expressed as a Percent
OBJECTIVE: To find percent change.
To find the relative error in linear and nonlinear measurements.
BELL RINGER: Use proportions to solve. a.) What percent of 240 is 60? b.) 90 is what percent of 120?
NOTES
A PERCENT CHANGE expresses an amount of change as a percent of an original amount.
If the new amount is greater than the original amount, the percent change is called a PERCENT
INCREASE.
If the new amount is less than the original amount, the percent change is called a PERCENT
DECREASE.
Percent Change is the ratio of the amount of change to the original amount.
Percent Change = amount of increase original or amount decrease

Can also be thought of as =
𝐶ℎ𝑎𝑛𝑔𝑒
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑥 100
Example 1: A car 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?
Example 2: The average monthly precipitation for Chicago, Illinois, peaks in June at 4.1 in. The average monthly precipitation in December is 2.8 in. What is the percent decrease from June to
December?
Example 3: A store buys an electric guitar for $295. The store then marks up the price of the guitar to
$340. What is the markup expressed as a percent change?
Example 4 : In one year, the toll for passenger cars to use a tunnel rose from $3 to $3.50. What was the percent increase?
RELATIVE ERROR is the ratio of the absolute value of the difference of a measured (or estimated) value and an actual value compared to the actual value. or estimated value
 actual
Relative Error = measured actual value value
When relative error is expressed as a percent, it is called PERCENT ERROR.
Example 5: A decorator estimates that a rectangular rug is 5 ft by 8 ft. The rug is actually 4 ft by 8 ft.
What is the percent error in the estimated area?

Example 6: You think that the distance between your house and your friend’s house is 5.5 mi. The actual distance is 4.75 mi. What is the percent error in your estimating?
Example 7:
A cube’s volume is estimated to be 8 cm 3 . When measured, each side was 2.1 cm in length. What is the percent error in the estimated volume?
Example 8: A student’s height is measured as 144 cm to the nearest centimeter. What are the student’s minimum and maximum possible heights?
Example 9: You 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?
Example 10: What is the greatest possible percent of error in calculating the volume for a cube with a side measured at 2.1 cm?
Example 11: The dimensions of a gift box to the nearest inch are 12 in x 6 in x 5 in. What is the greatest possible percent of error in calculating the volume of the gift box?