GLENCOE DIVISION · Answer: Twelve divided by b minus four equals ... Words The first height plus...

Algebra 1 Interactive Chalkboard

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GLENCOE DIVISION

Glencoe/McGraw-Hill

8787 Orion Place

Columbus, Ohio 43240

Lesson 3-1 Writing Equations

Lesson 3-2 Solving Equations by Using Addition

and Subtraction

Lesson 3-3 Solving Equations by Using Multiplication

and Division

Lesson 3-4 Solving Multi-Step Equations

Lesson 3-5 Solving Equations with the Variable

on Each Side

Lesson 3-6 Ratios and Proportions

Lesson 3-7 Percent of Change

Lesson 3-8 Solving Equations and Formulas

Lesson 3-9 Weighted Averages

Example 1 Translate Sentences into Equations

Example 2 Use the Four-Step Plan

Example 3 Write a Formula

Example 4 Translate Equations into Sentences

Example 5 Write a Problem

Translate this sentence into an equation.

A number b divided by three is equal to six less than c.

b divided by three is equal to six less than c.

Answer: The equation is .


Translate this sentence into an equation.

Fifteen more than z times six is y times two minus eleven.

Fifteen more than z times six is y times two minus eleven.

15 z 6 y 2 11



Translate each sentence into an equation.

a. A number c multiplied by six is equal to two more than d.

b. Three less than a number a divided by four is seven more than 3 times b.

Explore You know that 1,250,000 jellybeans are produced each hour. You want to know how many hours it will take to produce 10,000,000 jellybeans.

Jellybeans A popular jellybean manufacturer produces 1,250,000 jellybeans per hour. How many hours does it take them to produce 10,000,000 jellybeans?

Plan Write an equation to represent the situation. Let h represent the number of hours needed to produce the jellybeans.

1,2500,000 h 10,000,000

Solve

Find h mentally by asking, “What number times 125 equals 1000?”

Answer: It will take 8 hours to produce 10,000,000 jellybeans.

1,250,000 times hours equals 10,000,000.

h = 8

Examine If 1,250,000 jellybeans are produced in one hour, then 1,250,000 x 8 or 10,000,000 jellybeans are produced in 8 hours. The answer makes sense.

A person at the KeyTronic World Invitational

Answer: It would take 24 minutes.

Type-Off typed 148 words per minute. How

many minutes would it take to type 3552

words?

Answer: The formula is .

Translate the sentence into a formula.

Words Perimeter equals four times the length of the side.

Variables Let P = perimeter and s = length of a side.

The perimeter of a square equals four times the length of the side.

Perimeter equals four times the length of a side.

P 4s

Answer: The formula is .

Translate the sentence into a formula.

The area of a circle equals the product of and the square of the radius r.

Translate this equation into a verbal sentence.

Answer: Twelve minus two times x equals negative five.

Twelve minus two times x equals negative five.

12 2x 5

Translate this equation into a verbal sentence.

Answer: a squared plus three times b equals c divided by six.

a squared plus three times b equals c divided by six.

a2 3b

Translate each equation into a verbal sentence.

Answer: Twelve divided by b minus four equals negative one.

Answer: Five times a equals b squared plus one.

1.

2.

4( f + 1.50) – f = 8.25

Write a problem based on the given information.

Answer: The cost of a burger is $1.50 more than the cost of fries. Four times the cost of a burger minus the cost of fries equals $8.25. How much do fries cost?

f = cost of fries f + 1.50 = cost of a burger

h – 3 = Consuelo’s height in inches

3h(h – 3) = 8262

Write a problem based on the given information.

Answer: Consuelo is 3 inches shorter than Tiana. The product of Consuelo’s height and three times Tiana’s is 8262. How tall is Tiana?

h = Tiana’s height in inches

Example 1 Solve by Adding a Positive Number

Example 2 Solve by Adding a Negative Number

Example 3 Solve by Subtracting

Example 4 Solve by Adding or Subtracting

Example 5 Write and Solve an Equation

Example 6 Write an Equation to Solve a Problem

Solve . Then check your solution.

Original equation

Answer: and

To check that –15 is the solution, substitute –15 for h in the original equation.

Add 12 to each side.


Answer: 40

Check:


To check that 29 is the solution, substitute 29 for k in the original equation.

Original equation

Answer: and

Add –63 to each side.


Answer: –61

Check:


To check that –66 is the solution, substitute –66 for c in the original equation.

Original equation

Answer: and

Subtract 102 from each side.


Answer: –171

Check:

Solve in two ways.

Method 1 Use the Subtraction Property of Equality.

Original equation

and Answer:

or

Subtract from each side.

Method 2 Use the Addition Property of Equality.

Original equation

and Answer:

or

Add to each side.

Solve .

Answer:

Write an equation for the problem. Then solve the equation and check your solution.

Fourteen more than a number is equal to twenty-seven. Find this number.

Fourteen more than a number is equal to twenty-seven.

Original equation

14 n 27


To check that 13 is the solution, substitute 13 for n in the original equation. The solution is 13.

and Answer:

Twelve less than a number is equal to negative twenty-five. Find the number.

Answer: –13

History The Washington Monument in Washington, D.C., was built in two phases. From 1848–1854, the monument was built to a height of 152 feet. From 1854 until 1878, no work was done. Then from 1878 to 1888, the additional construction resulted in its final height of 555 feet. How much of the monument was added during the second construction phase?

Answer: There were 403 feet added to the Washington Monument from 1878 to 1888.

Words The first height plus the additional height equals 555 feet.

Variables Let a = the additional height.

The first height plus the additional height equals 555.

Original equation

and

152 a 555


The Sears Tower was built in 1974. The height to the Sky Deck is 1353 feet. The actual recorded height is 1450 feet. In 1982, they added twin antenna towers, which does not count for the record, for a total structure height of 1707 feet. How tall are the twin antenna towers?

Answer: 257 feet

Example 1 Solve Using Multiplication by a Positive

Number

Example 2 Solve Using Multiplication by a Fraction

Example 3 Solve Using Multiplication by a Negative

Number

Example 4 Write and Solve an Equation Using

Multiplication

Example 5 Solve Using Division by a Positive Number

Example 6 Solve Using Division by a Negative Number

Example 7 Write and Solve an Equation Using Division


To check that 9 is the solution, substitute 9 for s in the original equation.

Original equation

Answer: and

Multiply each side by 12.

Solve .

Answer: 12


Original equation

Rewrite each mixed number as in improper fraction.

Multiply each side by , the reciprocal of .

To check that is the solution, substitute for k in the original equation.

Answer: or Check this result.

Solve .

Answer:

Solve .

To check that 5 is the solution, substitute 5 for b in the original equation.

Original equation

Answer: Check this result.

Multiply each side by , the reciprocal of –15.

Solve .

Answer:

Space Travel Using information from Example 4 in the Student Edition, what would be the weight of Neil Armstrong’s suit and life-support backpack on Mars if three times the Mars weight equals the Earth weight?

Words Three times the weight on Mars equals the weight on Earth.

Variables Let w = the weight on Mars.

Three times the weight on Mars equals the weight on Earth.

Answer: The weight of Neil Armstrong’s suit and life-support backpacks on Mars would be 66 pounds.

Original equation

and

3 w 198

Multiply each side by .

Refer to the information about Neil Armstrong in Example 4. If Neil Armstrong weighed 216 pounds on Earth, how much would he weigh on Mars?

Answer: He would weigh 72 pounds on Mars.


Original equation

Answer: and

To check, substitute 13 for w.

Divide each side by 11.


Check:

Answer: 17

Solve .

Original equation

and Answer:

Divide each side by –8.

Solve .

Answer: –23

Write an equation for the problem below. Then solve the equation.

Negative fourteen times a number equals 224.

Original equation

Answer: Check this result.

Negative fourteen times a number equals 224.

n –14 224


Negative thirty-four times a number equals 578. Find the number.

Answer: –17

Example 1 Work Backward to Solve a Problem

Example 2 Solve Using Addition and Division

Example 3 Solve Using Subtraction and Multiplication

Example 4 Solve Using Multiplication and Addition

Example 5 Write and Solve a Multi-Step Equation

Example 6 Solve a Consecutive Integer Problem

Danny took some rope with him on his camping trip. He used 32 feet of rope to tie his canoe to a log on the shore. The next night, he used half of the remaining rope to secure his tent during a thunderstorm. On the last day, he used 7 feet as a fish stringer to keep the fish that he caught. After the camping trip, he had 9 feet left. How much rope did he have at the beginning of the camping trip?

Start at the end of the problem and undo each step.

Statement Undo the Statement

He had 9 feet left.

He used 7 feet as a fish stringer.

He used half of the remaining rope to secure his tent.

He used 32 feet to tie his canoe.

9

9 + 7 = 16

16 2 = 32

32 + 32 = 64

Answer: He had 64 feet of rope. Check the answer in the context of the problem.

Olivia went to the mall to spend some of her monthly allowance. She put $10 away so it could be deposited in the savings account at a later date. The first thing she bought was a CD for $15.99. The next stop was to buy hand lotion and a candle, which set her back $9.59. For lunch, she spent half of the remaining cash. She went to the arcade room and spent $5.00 and took home $1.21. How much was Olivia’s monthly allowance?

Answer: $48.00


Simplify.

Answer: Simplify.

To check, substitute 10 for q in the original equation.

Original equation



Solve .

Answer: 14


Simplify.

Original equation

Answer: s = –240 Simplify.

Multiply each side by 12.


To check, substitute –240 for s in the original equation.

Solve .

Answer: 363

Solve .

Original equation

Simplify.

Answer: r = 14 Simplify.

Multiply each side by –3.

Add 8 to each side.

Solve .

Answer: 21

Write an equation for the problem below. Then solve the equation.

Eight more than five times a number is negative 62.

Original equation

Simplify.

Eight more than five times a number is negative 62.

n 8 62 5


Simplify. Answer: n = –14

Multiply each side by .

Three-fourths of seven subtracted from a number is negative fifteen. What is the number?

Answer: –13

Number Theory Write an equation for the problem below. Then solve the equation and answer the problem.

Find three consecutive odd integers whose sum is 57.

Let n = the least odd integer.

Let n + 2 = the next greater odd integer.

Let n + 4 = the greatest of the three odd integers.

The sum of three consecutive odd integers is 57.

= 57

Simplify.

Original equation

Simplify.

Simplify.

or 19

or 21

Answer: The consecutive odd integers are 17, 19, and 21.



Find three consecutive even integers whose sum is 84.

Answer: 26, 28, 30

Example 1 Solve an Equation with Variables on Each Side

Example 2 Solve an Equation with Grouping Symbols

Example 3 No Solutions

Example 4 An Identity

Example 5 Use Substitution to Solve an Equation

Solve .

Original equation

Simplify.

Simplify.

Answer: s = 5 Simplify.

Subtract 7s from each side.



To check your answer, substitute 5 for s in the original equation.

Solve .

Answer:


Original equation

Distributive Property

Simplify.

Subtract 12q from each side.


Simplify.

Answer: q = 6 Simplify.

To check, substitute 6 for q in the original equation.


Solve .

Answer: 36

Solve .

Original equation


This statement is false.

Answer: There must be at least one c to represent the variable. This equation has no solution.

Subtract 40c from each side.

Solve .

Answer: This equation has no solution.


Original equation


Answer: Since the expression on each side of the equation is the same, this equation is an identity. The statement is true for all values of t.

Answer: is true for all values of c.

Solve .

Multiple-Choice Test Item

Solve .

A 13

B –13

C 26

D –26

Read the Test Item You are asked to solve an equation.

Solve the Test Item You can solve the equation or substitute each value into the equation and see if it makes the equation true. We will solve by substitution.

A: Substitute 13 for b.

B: Substitute –13 for b.

C: Substitute 26 for b.

D: Substitute –26 for b.

Answer: Since the value –26 makes the statement true, the answer is D.

Multiple-Choice Test Item

Solve .

A 32

B –32

C 26

D –26

Answer: C

Example 1 Determine Whether Ratios Form a Proportion

Example 2 Use Cross Products

Example 3 Solve a Proportion

Example 4 Use Rates

Example 5 Use a Scale Drawing

Answer: The ratios are equal. Therefore, they form a proportion.

Determine whether the ratios and form a proportion.

Do the ratios and form a proportion?

Answer: The ratios are not equal. Therefore, they do not form a proportion.

Use cross products to determine whether the pair of ratios below forms a proportion.

Write the equation.

Find the cross products.

Answer: The cross products are not equal, so . The ratios do not form a proportion.

Simplify.

Write the equation.


Answer: The cross products are equal, so . Since the ratios are equal, they form a proportion.


Simplify.


Answer: The cross products are equal. Therefore, the ratios do form a proportion.


Answer: The cross products are not equal. Therefore, the ratios do not form a proportion.

Solve the proportion .

Original equation


Simplify.


Answer: Simplify.

Solve the proportion .

Answer: 6.3

Explore Let p represent the number of times needed to crank the pedals.

Plan Write a proportion for the problem.

turns of the pedals

wheel turns

turns of the pedals

wheel turns

Bicycling The gear on a bicycle is 8:5. This means that for every eight turns of the pedals, the wheel turns five times. Suppose the bicycle wheel turns about 2435 times during a trip. How many times would you have to turn the pedals during the trip?

Solve Original proportion



Answer: 3896 = p Simplify.

Simplify.

Examine If it takes 8 turns of the pedal to make the wheel turn 5 times, then it would take 1.6 turns of the pedal to make the wheel turn 1 time. So, if the wheel turns 2435 times, then there are 2435 1.6 or 3896 turns of the pedal. The answer is correct.

Before 1980, Disney created animated movies using cels. These hand drawn cels (pictures) of the characters and scenery represented the action taking place, one step at a time. For the movie Snow White, it took 24 cels per second to have the characters move smoothly. The movie is around 42 minutes long. About how many cels were drawn to produce Snow White?

Answer: About 60,480 cels were drawn to produce Snow White.

Maps In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. The scale for the map of Texas is 5 inches = 144 miles. What are the distances in miles represented by 2.5 inches on each map?

Explore Let d represent the actual distance.

Plan Write a proportion for the problem.

scale

actual

scale

actual

Connecticut:

Texas:

scale

actual

scale

actual

Solve Connecticut:


Simplify.


Simplify. or 20.5

Solve Texas:


Simplify.


Simplify. or 72

Answer: The actual distance in Connecticut represented by 2.5 inches is 20.5 miles. The actual distance in Texas represented by 2.5 inches is 72 miles.

Examine: 2.5 inches is of 5 inches. So 2.5 inches

represents (41) or 20.5 miles in Connecticut

and (144) or 72 miles in Texas. The answer

is correct.

The scale on a map of the United States is inches = 750 miles. The distance, on the map, between Los Angeles and Washington, D.C., is about inches. What is the distance in miles between the two locations?

Answer: The distance in miles between Los Angeles and Washington, D.C., is about 2,114 miles.

Example 1 Find Percent of Change

Example 2 Find the Missing Value

Example 3 Find Amount After Sales Tax

Example 4 Find Amount After Discount

State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.

Find the amount of change. Since the new amount is greater than the original, the percent of change is a percent of increase.

40 – 32 = 8

original: 32 new: 40

Find the percent using the original number, 32, as the base.

change

original amount

percent change

100 percent


Simplify.

Simplify.

Answer: The percent of increase is 25%.


Find the amount of change. Since the new amount is less than the original, the percent of change is a percent of decrease.

20 – 4 = 16

State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.

original: 20 new: 4

Find the percent using the original number, 20, as the base.

change

original amount

percent change

100 percent


Simplify.

Simplify.

Answer: The percent of decrease is 80%.


a. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.

Answer: The percent of change is a decrease of 28%.


b. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.

Answer: The percent of change is an increase of 300%.


Sales The price a used-book store pays to buy a book is $5. The store sells the book for 28% above the price that it pays for the book. What is the selling price of the $5 book?

Let s = the selling price of the book. Since 28% is the percent of increase, the amount the used-book store pays to buy a book is less than the selling price. Therefore, s – 5 represents the amount of change.



Simplify.

Answer: The selling price of the $5 book is $6.40.

change

book store cost

percent change

100 percent

Simplify.



At one store the price of a pair of jeans is $26.00. At another store the same pair of jeans has a price that is 22% higher. What is the price of jeans at the second store?

Answer: The price of jeans at the second store is $31.72.

Sales Tax A meal for two at a restaurant costs $32.75. If the sales tax is 5%, what is the total price of the meal?

The tax is 5% of the price of the meal.

Use a calculator.

Round $1.6375 to $1.64. Add this amount to the original price.

Answer: The total price of the meal is $34.49.

A portable CD player costs $69.99. If the sales tax is 6.75%, what is the total price of the CD player?

Answer: The total price of the CD player is $74.71.

The discount is 20% of the original price.

Subtract $0.76 from the original price.

Answer: The discounted price of the dog toy is $3.04.

Discount A dog toy is on sale for 20% off the original price. If the original price of the toy is $3.80, what is the discounted price?

A baseball cap is on sale for 15% off the original price. If the original price of the cap is $19.99, what is the discounted price?

Answer: The discounted price of the cap player is $16.99.

Example 1 Solve an Equation for a Specific Variable

Example 2 Solve an Equation for a Specific Variable

Example 3 Use a Formula to Solve Problems

Example 4 Use Dimensional Analysis

Solve for b.

Original equation

Simplify.

Subtract 12c from each side.


or

Simplify.

Answer: The value of b is .

Solve for y.

Answer: The value of y is .

Solve for x.

Original equation

Simplify.

Simplify.

Use the Distributive Property.

Add xy to each side.

Add 2z to each side.

Answer: The value of x is . Since division by 0 is

undefined, .

Divide each side by 7 + y.

Answer: The value of a is .

Solve for a.

Fuel Economy A car’s fuel economy E (miles per gallon) is given by the formula , where m is the number of miles driven and g is the number of gallons of fuel used. Solve the formula for m.

Formula for fuel economy.

Simplify. Answer:

Multiply each side by g.

Fuel Economy If Claudia’s car has an average fuel consumption of 30 miles per gallon and she used 9.5 gallons, how far did she drive?

Formula for how many miles driven

E = 30 mpg and g = 9.5 gallons

Multiply.

Answer: She drove 285 miles.

Fuel Economy A car’s fuel economy E (miles per gallon) is given by the formula , where m is the number of miles driven and g is the number of gallons of fuel used. Solve the formula for g.

Answer:

If Claudia drove 1477 miles and her pickup has an average fuel consumption of 19 miles per gallon, how many gallons of fuel did she use?

Answer: She used around 77.74 gallons.

Geometry The formula for the volume of a cylinder is , where r is the radius of the cylinder and h is the height. Solve the formula for h.

Original formula

Answer:

Divide each side by .

Geometry What is the height of a cylindrical swimming pool that has a radius of 12 feet and a volume of 1810 cubic feet?

Answer: The height of the cylindrical swimming pool is about 4 feet.

Formula for h

V = 1810 and r = 12

Use a calculator.

Answer:

Geometry The formula for the volume of a cylinder is , where r is the radius of the cylinder and h is the height.

Solve the formula for r.

What is the radius of a cylindrical swimming pool if the volume is 2010 cubic feet and a height of 6 feet?

Answer: The radius is about 10 feet.

Example 1 Solve a Mixture Problem with Prices

Example 2 Solve a Mixture Problem with Percents

Example 3 Solve for Average Speed

Example 4 Solve a Problem Involving Speeds of

Two Vehicles

Pets Jeri likes to feed her cat gourmet cat food that costs $1.75 per pound. However, food at that price is too expensive so she combines it with cheaper cat food that costs $0.50 per pound. How many pounds of cheaper food should Jeri buy to go with 5 pounds of gourmet food, if she wants the price to be $1.00 per pound?

Units (lb) Price per Unit Price

Gourmet cat food

Gourmet cat food

Mixed cat food

Let w = the number of pounds of cheaper cat food. Make a table.

5 $1.75 $8.75

w $0.50 0.5w

5 + w $1.00 1.00(5 + w)

Original equation


Subtract 0.5w from each side.

Simplify.

Price of gourmet cat food

plus

price of cheaper cat food

equals

price of mixed

cat food.

Write and solve an equation using the information in the table.

8.75 0.5w 1.00(5 + w)

Simplify.

Answer: Jerry should buy 7.5 pounds of cheaper cat food to be mixed with the 4 pounds of gourmet cat food to equal out to $1.00 per pound of cat food.

Subtract 5.0 from each side.

Simplify.

Divide each side by 0.5.

Cheryl has bought 3 ounces of sequins that cost $1.79 an ounce. The seed beads cost $0.99 an ounce. How many ounces of seed beads can she buy if she only wants the beads to be $1.29 an ounce for her craft project?

Answer: Cheryl should buy 4.6 ounces of beads.

Auto Maintenance To provide protection against freezing, a car’s radiator should contain a solution of 50% antifreeze. Darryl has 2 gallons of a 35% antifreeze solution. How many gallons of 100% antifreeze should Darryl add to his solution to produce a solution of 50% antifreeze?

35% Solution

100% Solution

50% Solution

Amount of Solution (gallons)

Price

Let g = the number of gallons of 100% antifreeze to be added. Make a table.

2 0.35(2)

g 1.0(g)

2 + g 0.50(2 + g)

Original equation


Amount of antifreeze in 35% solution

plus

amount of antifreeze in

100% solution

equals

amount of antifreeze in

50% solution.


Simplify.

Subtract 0.50g from each side.

0.35(2) 1.0(g) 0.50(2 + g)

Subtract 0.70 from each side.

Simplify.

Answer: Darryl should add 0.60 gallons of 100% antifreeze to produce a 50% solution.

Simplify.

Divide each side by 0.50.

A recipe calls for mixed nuts with 50% peanuts. pound of 15% peanuts has already been used. How many pounds of 75% peanuts needs to be add to obtain the required 50% mix?

Answer: of a pound of 75% peanuts should be used.

To find the average speed for each leg of the trip, rewrite .

Air Travel Mirasol took a non-stop flight from Newark to Austin to visit her grandmother. The 1500-mile trip took three hours and 45 minutes. Because of bad weather, the return trip took four hours and 45 minutes. What was her average speed for the round trip?

Going

Returning

Simplify.

Answer: The average speed for the round trip was about 343.9 miles per hour.

Definition of weighted average

Round Trip

In the morning, when traffic is light, it takes 30 minutes to get to work. The trip is 15 miles through towns. In the afternoon when traffic is a little heavier, it takes 45 minutes. What is the average speed for the round trip?

Answer: The average speed for the round trip was about 23 miles per hour.

Rescue A railroad switching operator has discovered that two trains are heading toward each other on the same track. Currently, the trains are 53 miles apart. One train is traveling at 75 miles per hour and the other train is traveling at 40 miles per hour. The faster train will require 5 miles to stop safely, and the slower train will require 3 miles to stop safely. About how many minutes does the operator have to warn the train engineers to stop their trains?

Draw a diagram.

53 miles apart

Takes 5 miles to stop Takes 3 miles to stop

Fast train

Other train

r d = rt t

Let m = the number of minutes that the operator has to warn the train engineers to stop their trains safely. Make a table.

75 m

40 m

75m

40m

Distance traveled by fast train

plus

distance traveled by other train

equals

45 miles.


Original equation

Simplify.

75m 40m 45


Answer: The operator has about 23 minutes to warn the engineers.

Round to the nearest hundredth.

Convert to minutes by multiplying by 60.

Answer: They will be 7.5 miles apart in about 14 minutes.

Two students left the school on their bicycles at the same time, one heading north and the other heading south. The student heading north travels 15 miles per hour, and the one heading south travels at 17 miles per hour. About how many minutes will they be 7.5 miles apart?

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