7-1 Solving Two-Step Equations PRE-ALGEBRA LESSON 7-1 Solve each equation. a. 15 = 8 + n n = 7 b. p...

7-1

Solving Two-Step EquationsSolving Two-Step EquationsPRE-ALGEBRA LESSON 7-1PRE-ALGEBRA LESSON 7-1

Solve each equation.

a. 15 = 8 + n

n = 7

b. p – 19 = 4

p = 23



1. 9 + k = 17 2. d – 10 = 1 3. y – 5 = –4

4. x + 16 = 4 5. b + 6 = –4

(For help, go to Lesson 2-5.)

Check Skills You’ll Need

7-1


1. 9 + k = 17–9 + 9 + k = –9 + 17

k = 8

2. d – 10 = 1d – 10 + 10 = 1 + 10 d = 11

5. b + 6 = –4b + 6 – 6 = –4 – 6 b = –10

3. y – 5 = –4y – 5 + 5 = –4 + 5 y = 1

4. x + 16 = 4x + 16 – 16 = 4 – 16 x = –12

Solutions

7-1

Solve 5v – 12 = 8.


5v – 12 = 8

5v – 12 + 12 = 8 + 12 Add 12 to each side.

5v = 20 Simplify.

v = 4 Simplify.

= Divide each side by 5.5v5

205

Check 5v – 12 = 8

5(4) – 12 8 Replace v with 4.

20 – 12 8 Multiply.

8 = 8 Simplify. Quick Check

7-1

Solve 7 – 3b = 1.


7 – 3b = 1

–7 + 7 – 3b = –7 + 1 Add –7 to each side.

0 – 3b = –6 Simplify.

–3b = –6 0 – 3b = –3b.

b = 2 Simplify.

= Divide each side by –3.–3b–3

–6–3

Quick Check

7-1

You borrow $350 to buy a bicycle. You agree to

pay $100 the first week, and then $25 each week until the

balance is paid off. To find how many weeks w it will take

you to pay for the bicycle, solve 100 + 25w = 350.


It will take you 10 weeks to pay for the bicycle.

100 + 25w = 350

100 + 25w – 100 = 350 – 100 Subtract 100 from each side.

25w = 250 Simplify.

w = 10 Simplify.

= Divide each side by 25.25w25

25025

Quick Check

7-1



1. 12x – 14 = 10 2. + 7 = –4

3. 9 – w = 13 4. –22 – = –15

5. 4d – 57 = 7

r3

q5

2 –33

–4 –35

16

7-1

7-2

Solving Multi-Step EquationsSolving Multi-Step EquationsPRE-ALGEBRA LESSON 7-2PRE-ALGEBRA LESSON 7-2

Simplify each expression.

a. 2(18) + 3(21 ÷ 7)

45

b. 21 – 5 + 4x + 2(3 + x)

22 + 6x


Simplify each expression.

1. 2x + 4 + 3x 2. 5y + y 3. 8a – 5a

4. 2 – 4c + 5c 5. 4x + 3 – 2(5 + x)



7-2


Solutions

2. 5y + y = 5y + 1y= (5 + 1)y= 6y

1. 2x + 4 + 3x = (2 + 3)x + 4= 5x + 4

4. 2 – 4c + 5c = 2 + (–4c + 5c)= 2 + (–4 + 5)c= 2 + c

3. 8a – 5a = (8 – 5)a= 3a

5. 4x + 3 – 2(5 + x) = 4x + 3 – 10 – 2x= 4x – 2x – 7= (4 – 2)x – 7= 2x – 7

7-2

Solving Multi-Step EquationsSolving Multi-Step Equations

In his stamp collection, Jorge has five more than

three times as many stamps as Helen. Together they have

41 stamps. Solve the equation s + 3s + 5 = 41. Find the

number of stamps each one has.

PRE-ALGEBRA LESSON 7-2PRE-ALGEBRA LESSON 7-2

s + 3s + 5 = 41

4s + 5 = 41 Combine like terms.

4s + 5 – 5 = 41 – 5 Subtract 5 from each side.

4s = 36 Simplify.

s = 9 Simplify.

= Divide each side by 4.4s4

364

7-2


(continued)


Check Is the solution reasonable? Helen and Jorge have a total of 41 stamps. Since 9 + 32 = 41, the solution is reasonable.

Helen has 9 stamps. Jorge has 3(9) + 5 = 32 stamps.

Quick Check

7-2


The sum of three consecutive integers is 42. Find

the integers.


sum of three consecutive integers 42isWords

Let = the least integer.n

Then = the second integer,n + 1

and = the third integer.n + 2

+ +n n + 1 n + 2Equation 42=

7-2


(continued)


n + (n + 1) + (n + 2) = 42

(n + n + n) + (1 + 2) = 42 Use the Commutative and Associative Properties of Addition to group like terms together.

3n + 3 = 42 Combine like terms.

3n + 3 – 3 = 42 – 3 Subtract 3 from each side.

3n = 39 Simplify.

n = 13 Simplify.

= Divide each side by 3.3n3

393

7-2


(continued)


If n = 13, then n + 1 = 14, and n + 2 = 15. The three integers are 13, 14, and 15.

Check Is the solution reasonable? Yes, because 13 + 14 + 15 = 42.

Quick Check

7-2




a. 4(2q – 7) = –4

4(2q – 7) = –4

8q – 28 = –4 Use the Distributive Property.

8q – 28 + 28 = –4 + 28 Add 28 to each side.

8q = 24 Simplify.

q = 3 Simplify.

Divide each side by 8.=8q8

248

7-2


(continued)


b. 44 = –5(r – 4) – r

44 = –5(r – 4) – r

44 = –5r + 20 – r Use the Distributive Property.

44 = –6r + 20 Combine like terms.

44 – 20 = –6r + 20 – 20 Subtract 20 from each side.

24 = –6r Simplify.

–4 = r Simplify.

Divide each side by –6.=24–6

–6r–6

Quick Check

7-2



1. b + 2b – 11 = 88 2. 6(2n – 5) = –90 3. 3(x + 6) + x = 86

4. Find four consecutive integers whose sum is –38.

33 –5 17

–11, –10, –9, –8

7-2

7-3

Multi-Step Equations With Fractions and DecimalsMulti-Step Equations With Fractions and DecimalsPRE-ALGEBRA LESSON 7-3PRE-ALGEBRA LESSON 7-3

Evaluate each algebraic expression.

a. c + 5 • 2 for c = 7 17

b. 20 – 8 ÷ p for p = 2 16



1. 2n + 53 = 47 2. 4m – 37 = –28

3. –26x – 4 = 100 4. –3a + 15 = 13



7-3


Solutions

1. 2n + 53 = 47 2. 4m – 37 = –282n + 53 – 53 = 47 – 53 4m – 37 + 37 = –28 + 37 2n = –6 4m = 9

n = –3 m = 2

3. –26x – 4 = 100 4. –3a + 15 = 13–26x – 4 + 4 = 100 + 4 –3a + 15 – 15 = 13 – 15 –26x = 104

x = –4 a =

=4m4 =

=–3a–3 =

–2–3

2n2

–62

–26x–26

104–26

23

9414

7-3

Multi-Step Equations With Fractions and DecimalsMulti-Step Equations With Fractions and Decimals

Solve p – 7 = 11.


p – 7 = 1134

Add 7 to each side.p – 7 + 7 = 11 + 734

Simplify.p = 1834

p =34

43 •

43 • 18 Multiply each side by , the reciprocal of .

34

43

1p =4 • 18

3 1

6Divide common factors.

p = 24 Simplify.

34

7-3


(continued)


Check p – 7 = 11

(24) – 7 11 Replace p with 24.

Simplify.18 – 7 11

11 = 11

34

34

– 7 11 Divide common factors.3 • 24

4 1

6

Quick Check

7-3


Solve y + 3 = .


y + 3 = 12

23

Multiply each side by 6, the LCM of 2 and 3.6 y + 3 = 12 6

23

3y + 18 – 18 = 4 – 18 Subtract 18 from each side.

3y = –14 Simplify.

3y + 18 = 4 Simplify.

Use the Distributive Property.6 • y + 6 • 3 =12 6

23

Divide each side by 3.3y3

–143=

Simplify.y = –423

23

12

Quick Check

7-3


Suppose your cell phone plan is $30 per month plus

$.05 per minute. Your bill is $36.75. Use the equation 30 +

0.05x = 36.75 to find the number of minutes on your bill.

There are 135 minutes on your bill.

30 + 0.05x = 36.75

x = 135 Simplify.

30 – 30 + 0.05x = 36.75 – 30 Subtract 30 from each side.

0.05x = 6.75 Simplify.

Divide each side by 0.05.=6.750.05

0.05x0.05

Quick Check

7-3



1. d + 13 = 20 2. (s – 8) = 9 3. 5 + = 14

4. 0.07x + 0.03 = 0.38 5. 0.015w – 1.85 = 1.615

16

35

2c4

42 23 18

5 231

7-3

7-4

Problem Solving Strategy: Write an EquationProblem Solving Strategy: Write an EquationPRE-ALGEBRA LESSON 7-4PRE-ALGEBRA LESSON 7-4

Write each phrase as an algebraic expression.

a. 12 times a number

12n

b. 8 less than a number

n – 8

c. twice the sum of 5 and a number

2(5 + n)

Problem Solving Strategy: Write an Equation Problem Solving Strategy: Write an Equation PRE-ALGEBRA LESSON 7-4PRE-ALGEBRA LESSON 7-4

Write an equation to represent each situation.

1. Pierre bought a puppy for $48. This is $21 less than the original price. What was the original price of the puppy?

2. A tent weighs 6 lb. Together, your backpack and the tent weigh 33 lb. How much does your backpack weigh?

3. A veterinarian weighs 140 lb. She steps on a scale while holding a large dog. The scale shows 192 lb. What is the weight of the dog?



7-4


Solutions

1. Price of puppy minus $21 is $48.Let p = the original price of the puppy.p – 21 = 48

2. Weight of backpack plus weight of tent is 33 lb.Let b = the weight of the backpack.b + 6 = 33

3. The weight of the veterinarian plus the weight of the dog is 192 lb.Let d = the weight of the dog.140 + d = 192

7-4

Problem Solving Strategy: Write an EquationProblem Solving Strategy: Write an Equation

A moving van rents for $29.95 a day plus $.12 a

mile. Mr. Reynolds’s bill was $137.80 and he drove the van

150 mi. For how many days did he have the van?


Words number of days • $29.95/d + $.12/mi • 150 mi $137.80=

Let d = number of days Mr. Reynolds had the van.

Equation 137.80d • 29.95 + 0.12 • 150 =

7-4

Problem Solving Strategy: Write an EquationProblem Solving Strategy: Write an Equation

(continued)


Mr. Reynolds had the van for 4 days.

d • 29.95 + 0.12 • 150 = 137.80

29.95d + 18 = 137.80 Multiply 0.12 and 150.

29.95d + 18 – 18 = 137.80 – 18 Subtract 18 from each side.

29.95d = 119.80 Simplify.

d = 4 Simplify.

Divide each side by 29.95.=29.95d29.95

119.8029.95

Quick Check

7-4


Write an equation. Then solve.

1. You buy 2 pounds of sliced roast beef for $3 per pound and some smoked turkey for $5 per pound. You spend $13.50. How much turkey did you buy?

2. A phone call costs $.35 for the first minute and $.15 for each additional minute. The phone call lasted 14 minutes. How much did the call cost?

$2.30

1.5 lb

7-4

7-5

Solving Equations With Variables on Both SidesSolving Equations With Variables on Both SidesPRE-ALGEBRA LESSON 7-5PRE-ALGEBRA LESSON 7-5

On Jim’s vacation, he collected the same number of shells each day for 7 days. When he came home, he gave away 14 shells and had 28 left over. How many shells did he collect each day of his vacation?

6 shells



1. k + 3k = 20 2. 8x – 3x = 35

3. 3b + 2 – b = –18 4. –8 – y + 7y = 40



7-5


Solutions

1. k + 3k = 20 2. 8x – 3x = 35 4k = 20 5x = 35 k = 5 x = 7

3. 3b + 2 – b = –18 4. –8 – y + 7y = 40 (3b – b) + 2 = –18 –8 + 6y = 40 2b + 2 = –18 –8 + 8 + 6y = 40 + 8 2b + 2 – 2 = –18 – 2 6y = 48 2b = –20

y = 8 b = –10

= =

==

204

4k4

355

5x5

486

6y6–20

22b2

7-5

Solving Equations With Variables on Both SidesSolving Equations With Variables on Both Sides

Solve 4c + 3 = 15 – 2c.


4c + 3 = 15 – 2c

4c + 2c + 3 = 15 – 2c + 2c Add 2c to each side.

6c + 3 = 15 Combine like terms.

6c + 3 – 3 = 15 – 3 Subtract 3 from each side.

6c = 12 Simplify.

c = 2 Simplify.

Divide each side by 6.=6c6

126

Check 4c + 3 = 15 – 2c4(2) + 3 15 – 2(2) 8 + 3 15 – 4 11 = 11

Substitute 2 for c.Multiply. Quick Check

7-5


Steve types at a rate of 15 words/min and Jenny types at a

rate of 20 words/min. Steve and Jenny are both typing the same

document, and Steve starts 5 min before Jenny. How long will it take

Jenny to catch up with Steve?

words Jenny types = words Steve types

Words 20 words/min • Jenny’s time = 15 words/min • Steve’s time

Let = Jenny’s time.x

Then x + 5 = Steve’s time.

Equation 20 • x = 15 • (x + 5)

7-5


(continued)

20x = 15(x + 5)

Jenny will catch up with Steve in 15 min.

20x = 15x + 75 Use the Distributive Property.

20x – 15x = 15x – 15x + 75 Subtract 15x from each side.

5x = 75 Combine like terms.

x = 15 Simplify.

Divide each side by 5.=5x5

755

7-5


(continued)

Check Test the result.At 20 words/min for 15 min, Jenny types 300 words.Steve’s time is five min longer. He types for 20 min.At 15 words/min for 20 min, Steve types 300 words.Since Jenny and Steve each type 300 words, the answer checks.

Quick Check

7-5



1. 3 – 2t = 7t + 4 2. 18 + 6z = 4z

3. 2q – 4 = 5 + 5q 4. 7(v – 4) = 3(3 + v) – 1

5. You work for a delivery service. With Plan A, you can earn $5 per hour plus $.75 per delivery. With Plan B, you can earn $7 per hour plus $.25 per delivery. How many deliveries must you make per

hour to earn the same amount from either plan?

19

– –9

–3 9

4 deliveries

7-5

7-6

Solving Two-Step InequalitiesSolving Two-Step InequalitiesPRE-ALGEBRA LESSON 7-6PRE-ALGEBRA LESSON 7-6

What number is 42,625 less than the sum of 62,345 and 51,284?

71,004


Solve each inequality. Graph the solutions.

1. w + 4 –5 2. 7 < z – 3

3. 4 > a + 6 4. x – 5 –6

>

<



7-6


Solutions

1. w + 4 –5 2. 7 < z – 3w + 4 – 4 –5 – 4 7 + 3 < z – 3 + 3 w –9 10 < z

z > 10

>>

>

3. 4 > a + 6 4. x – 5 –64 – 6 > a + 6 – 6 x – 5 + 5 –6 + 5 –2 > a x –1 a < –2

<<<

7-6

Solving Two-Step InequalitiesSolving Two-Step Inequalities

Solve and graph 7g + 11 > 67.


7g + 11 > 67

7g + 11 – 11 > 67 – 11 Subtract 11 from each side.

7g > 56 Simplify.

g > 8 Simplify.

Divide each side by 7.>7g7

567

Quick Check

7-6


Solve 6 – r – 6.


< 23

6 – r – 6<23

6 + 6 – r – 6 + 6< 23

Add 6 to each side.

Simplify.12 – r< 23

Simplify.>–18 r, or r –18<

32– (12)

32–

23– r Multiply each side by . Reverse the direction of

the inequality symbol.

32

–>

Quick Check

7-6


Dale has $25 to spend at a carnival. If the admission to

the carnival is $4 and the rides cost $1.50 each, what is the

greatest number of rides Dale can go on?


25Inequality 4 + •1.5 r <

Let = number of rides Dale goes on.

Words $4 admission + $1.50/ridenumberof rides

is less thanor equal to

$25•

r

7-6


(continued)


The greatest number of rides Dale can go on is 14.

4 + 1.5r 25<

Subtract 4 from each side.4 + 1.5r – 4 25 – 4<

Simplify.1.5r 21<

Divide each side by 1.5.1.5r1.5

211.5<

Simplify.r 14<

Quick Check

7-6


Solve each inequality.

1. 14 > 4d – 10 2. + 8 7

3. 32 – 12g < 176 4. 8 + 5a 23 >

<

d < 6 k 3>

g > –12 a 3>

–k3

7-6

7-7

Transforming FormulasTransforming FormulasPRE-ALGEBRA LESSON 7-7PRE-ALGEBRA LESSON 7-7

How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft)

12


Use each formula for the values given.

1. Use the formula d = rt to find d when r = 80 km/h and t = 4 h.

2. Use the formula P = 2 + 2w to find P when = 9 m and w = 7 m.

3. Use the formula A = bh to find A when b = 12 ft and h = 8 ft.


12


7-7


Solutions

2. P = 2 + 2wP = 2(9) + 2(7)P = 18 + 14P = 32 m

1. d = rtd = (80)(4)d = 320 km

3. A = bh

A = (12)(8)

A = (96)

A = 48 ft2

12

12

12

7-7

Solve the circumference formula C = 2 r for r.


C = 2 r

Use the Division Property of Equality.C

2 2 r2 =

Simplify.C

2 = r, or r =C

2

Quick Check

7-7

Transforming FormulasTransforming Formulas

Solve the perimeter formula P = 2 + 2w for w.


P = 2 + 2w

Simplify.P – 2 = 2w

Subtract 2 from each side.P – 2 = 2 + 2w – 2

Multiply each side by .12

12 (P – 2 ) = (2w)

12

12 P – = w Use the Distributive Property and simplify.

Quick Check

7-7


You plan a 600-mi trip to New York City. You estimate

your trip will take about 10 hours. To estimate your average

speed, solve the distance formula d = rt for r. Then substitute to

find the average speed.


d = rt

Your average speed will be about 60 mi/h.

= 60 Simplify.

Divide each side by t.=dt

rtt

Simplify.= r, or r =dt

dt

Replace d with 600 and t with 10. r =60010

Quick Check

7-7


The high temperature one day in San Diego was 32°C.

Solve C = (F – 32) for F. Then substitute to find the

temperature in degrees Fahrenheit.


59

C = (F – 32)59

(C) = (F – 32)95

95

59

Multiply each side by .95

Simplify.C = F – 3295

7-7


(continued)


32°C is 89.6°F.

Add 32 to each side.C + 32 = F – 32 + 3295

Simplify and rewrite.C + 32 = F, or F = C + 3295

95

Replace C with 32. Simplify.F = (32) + 32 = 89.695

Quick Check

7-7


Solve for the given variable.

1. Solve the area formula A = bh for b.

2. Solve the averaging formula a = for c.

3. The speed v of a satellite as it orbits Earth may be found using

the formula v2 = . Solve this formula for m, the mass of Earth.

b =2Ah

c = 2a – b

v2rGm =

12

b + c2

Gmr

7-7

7-8

Simple and Compound InterestSimple and Compound InterestPRE-ALGEBRA LESSON 7-8PRE-ALGEBRA LESSON 7-8

Use your classmates as subjects and estimate these percents.

a. About what percent are girls?

b. About what percent are twins?

Check students’ answers.



Find each amount.

1. 6% of $400 2. 55% of $2,000

3. 4.5% of $700 4. 5 % of $32512


7-8


1. 2. 6(400) = 100x 55(2000) = 100x

$24 = x $1,100 = x

3. 4. 4.5(700) = 100x 5.5(325) = 100x

$31.50 = x $17.88 = x

6100

x400=

55100

x2000=

4.5100

x700=

5.5100

x325=

x100=

6(400)100 =

100x100

55(2000)100

100x100

=5.5(325)

100100x100=

4.5(700)100

Solutions

7-8

Simple and Compound InterestSimple and Compound Interest

Suppose you deposit $1,000 in a savings account

that earns 6% per year.


a. Find the interest earned in two years. Find the total of principal plus interest.

The account will earn $120 in two years. The total of principal plus interest will be $1,120.

I = prt Use the simple interest formula.

I = 1,000 • 0.06 • 2 Replace p with 1,000, r with 0.06, and t with 2.

I = 120 Simplify.

total = 1,000 + 120 = 1,120 Find the total.

7-8


(continued)


b. Find the interest earned in six months. Find the total of principal plus interest.

The account will earn $30 in six months. The total of principal plus interest will be $1,030.

I = prt Use the simple interest formula.

I = 1,000 • 0.06 • 0.5 Replace p with 1,000, r with 0.06, and t with 0.5.

I = 30 Simplify.

Total = 1,000 + 30 = 1,030 Find the total.

Write the months as part of a year.t = = = 0.512

612

Quick Check

7-8

Year 5 : $486.20 486.20 • 0.05 = 24.31 486.20 + 24.31 = 510.51

Year 6 : $510.51 510.51 • 0.05 25.53 510.51 + 25.53 = 536.04

Year 7 : $536.04 536.04 • 0.05 26.80 536.04 + 26.80 = 562.84

Year 8 : $562.84 562.84 • 0.05 28.14 562.84 + 28.14 = 590.98


You deposit $400 in an account that earns 5% interest compounded annually (once per year). The balance after the first four years is $486.20. What is the balance in your account after another 4 years, a total of 8 years? Round to the nearest cent.


After the next four years, for a total of 8 years, the balance is $590.98.

Interest BalancePrincipal at

Beginning of Year

Quick Check

7-8


Find the balance on a deposit of $2,500 that earns 3%

interest compounded semiannually for 4 years.


The interest rate r for compounding semiannually is 0.03 ÷ 2, or 0.015.

The number of payment periods n is 4 years 2 interest periods per year, or 8.

The balance is $2,816.23.

B = p(1 + r)n Use the compound interest formula.

B = 2,500(1 + 0.015)8 Replace p with 2,500, r with 0.015, and n with 8.

Use a calculator. Round to the nearest cent.B 2,816.23

Quick Check

7-8


Find the simple interest and the balance.

1. $1,200 at 5.5% for 2 years 2. $2,500 at 8% for 6 months

3. Find the balance on a deposit of $1,200, earning 9.5% interest compounded semiannually for 10 years. $3,035.72

$132; $1,332 $100; $2,600

7-8

7-1 Solving Two-Step Equations PRE-ALGEBRA LESSON 7-1 Solve each equation. a. 15 = 8 + n n = 7 b. p...

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Transcript of 7-1 Solving Two-Step Equations PRE-ALGEBRA LESSON 7-1 Solve each equation. a. 15 = 8 + n n = 7 b. p...