3.1 Solving Equations Using Addition & Subtraction

Objectives:1. To solve linear equations using addition and subtraction2. To use linear equations to solve word problems involving real-world situations

Solving Linear Equations Using Addition

In this strategy, we balance the sides of the equation as we solve for the variable.

To solve equations like , you can use mental math.Ask yourself, “What number minus 3 is equivalent to 6?”

63 x

This strategy can work for easier problems, but we need a better plan so we can solve more difficult problems.

Since the “6” is the quantity after x subtract “3”. So, first, we must “undo” the minus 3. The inverse (opposite) of subtracting 3 is _________________ 3.

63 x

But we have to be fair! If we are going to add 3 to one side of the equation, we MUST add 3 to the other side to BALANCE the equation.9x

x

3 363x

add

Don’t forget to check your answer!!!

Solving Linear Equations Using Addition

Examples:

1) 124 x 2) 10655 x

3) 614 x 4) x 79

We must isolate the variable. What number is on the same side of the = with the variable? How do you undo this operation?

How can you check your answers?

4 44 12x

16x

55 5555 106x

51x

6 614 6x

20 x

7 79 7 x

2 x

Solving Linear Equations Using Subtraction

52x

First, we must “undo” the plus 2. The inverse (opposite) of adding 2 is ______________ 2.52 x

But we have to be fair! If we are going to subtract 2 from one side of the equation, we MUST subtract 2 from the other side to BALANCE the equation.3x

The process for solving equations using subtraction is very similar to solving equations using addition.How do you undo addition?

Example:

- 2 - 2

97x)1Ex 1015x)2Ex 3c11)3Ex

subtract

7– 7–7 9x

2x

15–

15–

15 10x

25x

3– 3–11 3c

8 c

Simplifying First

What needs to be done before you can solve these??

1) 12)2( a 2) 372 x

3) 29)(4 x 5.12)(5 y4)

2– 2–2 12a

10a

2+ 2+2 4x

6x

4 18x

14x

4– 4– 5 0.5y

5.5y

5+ 5+5 0.5y

Negative Signs with VariablesIn some problems, you will see a negative sign in front of the variable you are solving for.

Example: 12 x What does this mean?

The negative sign means “the opposite of x,” meaning the opposite sign, positive or negative.

12x x = -12

Another way is that you imagine you are the variable and the negative sign is the image.

Thinking of that you are standing in front of a mirror (the zero), where is your image? Or the reverse question: if you know where is your image, where are you standing?

Think the variable “x” is yourself and then the “–x” is your image in the mirror.

0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1-4-5-6-7-8-9-10

Negative Signs with Variables

To solve, we change the sign of the other side of the equation.

–x = –5

72 x4 2

15 5z 7)(6 x

Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54 degrees Fahrenheit at 9:00 am to – 4 degrees Fahrenheit at 9:27 am. By how many degrees did the temperature fall?

You started with some money in your pocket. All you spent was $4.65 on lunch. You ended up with $7.39 in your pocket. Write an equation to find out how much money you started with.

Word Problems - SWEET!

54 4x 58x

54– 54– 58x

4.65 7.39x

12.04x

4.65+ 4.65+

Summary

1. Inverse operations are operations that undo each other, such as “+” and “–”, “ · ” and “”.

2. You must apply the inverse operation at both sides of the equation at the same time. So the equation is kept in balance.

3. When the variable has a negative sign, you can imagine you are the variable and the negative sign represents the image.

3.2 Solving Equations Using Multiplication

To solve equations like , you will need to use multiplication.

24

x

24

x

In this problem, the x is being divided by 4. To solve for x, we will need to do the inverse of dividing by 4, which is multiplying by 4. **Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!

244

4

x

8x

Ex) 102

a

102

a

Objectives: To solve linear equations using multiplication and division and to use linear equations to solve word problems involving real-world situations

102

a( 2) ( 2) 20a

Don’t forget to check your answer!

Ex) 36 x

Solving Equations Using Division

To solve equations like , we can use division.324 x

324 x In this problem, the x is being multiplied by 4. To solve for x, we will need to do the inverse of multiplying by 4, which is DIVIDING by 4. **Don’t forget that you will need to do this to BOTH sides of the equation to keep it balanced!

4

32

4

4x

8x–6 –6

1

2x

You try it! Remember – locate the variable and undo whatever operation is being done to it! Also, simplify before you start!

35.1

x)1

43c

)5 7

y

31

)4

21x4)2 x102)3

)b5(1.8)6

4.5x 21

4x 1

5x

7

3x

43

c

12c

8.1 5b 8.1

5b

1.62b

Fraction Time

83

2x

How do we solve an equation like this?

Right now, the x is being multiplied by .The inverse of multiplying by is dividing by .

3

2

3

2

3

2

328

32

32

x

2

3

1

8x

12x

How do we divide by fractions? Multiply by the RECIPROCAL!

3

2 8

3

2x

Fraction Time

83

2x

How do we solve an equation like this?

Right now, the x is being multiplied by .The inverse of multiplying by is dividing by .

3

2

3

2

3

2

2

3

1

8x

12x

Ex)

7

12

5

1 y

How do we divide by fractions? Multiply by the RECIPROCAL!

3

2

1 12

5 7y

5 5

1 1

60

7y

Ex) It takes 16 cans of chili to make 4 batch of George’s extra-special chili-cheese dip. How many cans does it take to make 3 and a half batches? Write an equation and solve it. Be sure to state what your variables represent.

Ex) Amy’s mom has made 72 cookies to the members of Amy’s soccer team. If each of the team members (including 15 players, 2 coaches, and a manager) receives same number of cookies with no extra cookies left, how many cookies did Amy’s team member receive? Write an equation an solve it.

Can you solve equations involving multiplication and division?

Suppose we need x cans of chili to make 1 batch of chili-cheese dip, then

4 x = 16

x = 4

(3.5) · x = (3.5) · 4 = 14

Suppose each team member received n cookies, then

18 n = 72 n = 4

Summary

1. Always keep BOTH sides of the equation balanced when solving equation using multiplication and division.

2. If the number in front of the variable is a whole number, simply divide this number at BOTH sides of the equation.

3. If the number in front of the variable is a fraction, it is better to multiply the reciprocal of the this fraction at BOTH sides of the equation.

You will be asked to solve problems like which require more than one step to solve.

Guidelines for Solving Multi-Step Equations: 1) Simplify both sides if necessary distribute & combine like terms 2) To solve you must undo the order of operations BACKWARDS!

Start with Undo add or subtractThen Undo multiplication and division

713 x

713 x1 1

63 x3 3

2x

There is nothing to simplify What goes first?Simplify to get a new equation.

What goes next?

3.3 Multi-Step EquationsObjectives: To use 2 or more steps to solve a linear equation and to use multi-step equations to solve word problems

How can you check you answer?You MUST show all work on assignments, quizzes and tests.

Examples: Remember Do you need to simplify before you start.

1) 423 x 53)12(2 x2)

3) 83724 xx 4) 8 3( 2) 21x x

3 6x +2 +2

3 6

3 3

x

2x

4 2 3 5x 4 5 5x

–5 –54 10x

4 10

4 4

x

5

2x

24 4 8x +8 +8

32 4x32 4

4 4

x 8 x

8 3 6 21x x 5 6 21x

–6 –6

5 15x

5 15

5 5

x

3x

1124)2(3 xx13463 xx

1367 x6 6

77 x7 7

1x

simplifyDistribute

Combine like terms

Lots of steps! Where should you start?? 1124)2(3 xx

39

4x

4 3 4( 9)

3 4 3x

12x

Simplify

Subtract 3

Combine constant like terms

Lots of steps! Where should you start?? 3

3 64x

3 3

Multiply by the reciprocal of – 3/4

–3

Multiplying by a Reciprocal First

Given the example , what would you do first? 853

2x

873

4x 3

5

666 x

Let’s try something that may make this simpler!! Undo the first. HOW? 3

2

3 4 3( 7) 8

4 3 4x

2

7 6x

13x +7 +7

63 66

5x

5 6 5( 3) 66

6 5 6x

11

3 55x

58x –3 –3

Ex) The bill (parts and labor) for the repair of a car was $400. The cost of parts was $150, and the labor cost was $50 per hour. Write and solve and equation to find the number of hours of labor.

(HINT: this would make a GREAT quiz question!!! )

Let the number of hours of labor be x, then the labor cost is 50x and the parts cost is $150. The total cost $400 is the parts cost and labor cost. So

50 x + 150 = 400

–150 –150

50 x = 250

50 50

x = 5

Summary

1. When solving a multiple step equation, simplify both sides before you start. You may use distributive property, combine like terms, or multiply a fraction.

2. To solve you must undo the order of operations BACKWARDS! (inverse operation)Start with Undo add or subtract (+ or – )Then Undo multiplication and division ( · or )

3.4 Solving Equations with Variables on Both SidesObjective: To solve equations with variables on both sides of the equation.

Warm-up:

105x1552

)216)5x(4x2)1

How would you rate yourself on solving these How would you rate yourself on solving these problems?problems?

GREAT!! OK – getting there Need some help, but ok Need to come in for help

2x – 6x + 20 = –16

–4x + 20 = –16

x = 9

6x – 2 = –10

6x = –8

x = –4/3 5 2 5

(15 5) 102 5 2

x –5

15 5 25x 15 20x +5 +5 4 / 3x

–4x + 20 = –16 –20 –20

–4x = –36

1st: Variables to one side How do you decide who How do you decide who to move?to move?2nd: Constants to the other side Who must you Who must you move?move?

7x + 19 = -2x – 17

x 4

1) Which side has the smaller coefficient?2) Add 2x to both sides.

3) Simplify.9x + 19 = –174) Subtract 19 from both sides.9x = –36

6) Divide both sides by 9.

5) Simplify.

+2x +2x

–19 –19

9 36x 9 9 7) Simplify.

Can you check you answer? How?

1st: Variables to one side How do you decide who How do you decide who to move?to move?2nd: Constants to the other side Who must you Who must you move?move?

6x + 22 = -3x + 31

x 1

1) Which side has the smaller coefficient?2) Add 3x to both sides.

3) Simplify.9x + 22 = 314) Subtract 22 from both sides.9x = 9

6) Divide both sides by 9.

5) Simplify.

+3x +3x

–22 –22

9 9x 9 9 7) Simplify.

Can you check you answer? How?

1) 80 – 9y = 6y

Let’s try some!!

)2x(31016x1241

)4 3) 4(1 – x) + 3x = –2(x + 1)

How is this one How is this one different?different?

2) 64 – 12 w = 6w+9y +9y

80 15y80 15

15 15

y

16

3y

+12w +12w

64 18w64 18

18 18

y

32

9y

4 – 4x + 3x = –2x – 24 – x = –2x – 2+2x +2x4 + x = – 2–4 –4x = – 6

3x + 4 = 10 – 3x + 63x + 4 = 16 – 3x

+3x +3x6x + 4 = 16

–4 –46x = 12 x = 2

1st: Distribution How do you decide? Why?How do you decide? Why?

4(1 – x) + 3x = –2(x + 1)

3) Add 2x to both sides.

4) Simplify. x + 4 = –25) Subtract 4 from both sides. x = –6 6) Simplify.

+2x +2x

–4 –4

Can you check you answer? How?

1) Distribution.–x + 4 = –2x –

2

2) Simplify.4 – 4x + 3x = –2x – 2

Ex.

1st: Distribution How do you decide? Why?

112 16 10 3( 2)

4x x

3 4 10 3 6x x

3) Add 3x to both sides.

4) Simplify.

5) Subtract 4 from both sides.

8) Simplify.

1) Distribution.2) Simplify.3 4 16 3x x

6 4 16x +3x +3x

–4 –4

x 2

6 12x 6 6 7) Divide 6.

6) Simplify.

More Examples…You try these!!

)3x(41815x1052

)6 5) 9x + 22 = –3x + 46

12x + 22 = 46+3x +3x

–22 –22 12x = 24

x 2 12 12

4x + 6 = 18 – 4x + 12

+4x +4x8x + 6 = 30

4x + 6 = 30 – 4x

–6 –6 8x = 24

x 3 8 8

2 More of the fun type!!

7) 2(3 – 2x) + x = –3(x + 1)

8) 3(x – 5) – 6 = -21 + 3x

6 – 4x + x = –3x – 3

6 – 3x = –3x – 3 +3x +3x

6 = – 3

This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write

inconsistent

and conclude that

No solution

3x – 15 – 6 = –21 – 3x– 3x – 21 = –21 – 3x +3x +3x

–21 = –21

This kind of equation is called “identity equation”. When you get an identity equation, you just write

identity

and conclude thatMany solutions

2 More questions of the fun type!!

9) -4(1+2x) - 2x = –5(2x+1)

10) 2(3x – 1) – 6 = -8 + 6x

– 4 – 8x – 2x = –10x – 5–4 – 10x = –10x – 5 +10x +10x

–4 = – 5

This kind of equation is called “inconsistent equation”. When you get an inconsistent equation, you just write

inconsistent

and conclude that

No solution

6x – 2 – 6 = –8 + 6x

6x – 8 = –8 + 6x –6x –6x

–8 = –8

This kind of equation is called “identity equation”. When you get an identity equation, you just write

identity

and conclude thatMany solutions

Summary

1. When solving the equations with variables at both sides, you need apply the distributive property, combining like terms, and other skills.

2. Always simplify the equation first.3. Always work on the variable before to work

on the constant. Move the variables to one side and the constants to the other side.

4. Moving the variable term with less coefficient to the other side that the variable term with larger coefficient.

5. Be aware of two extreme situations: a) No solution you meet an inconsistent

equation when solving.b) Many solutions you meet an identity

equation when solving.

Assignment

P 157 #’s 18 - 44

3.5 Linear Equations and Problem Solving

Word Problems!!!My Favorite

Keys to succeed!

Write down important information

Draw a picture

Put the info in a chart if you can

Def

ine

your

var

iabl

e!!

We will meet the following typical types of real-life application questions:

1. Consecutive Integers2. Geometry3. Traveling4. Tickets5. Accounting

The real-life application questions are not just limited to the those above types. Others may be:

6. Clock (combination of Geometry and Traveling)7. Work8. Mixture

1) Find three consecutive integers whose sum is 162.

Consecutive Integers

Integer 1 x - 1Integer 2 xInteger 3 x + 1Total 162

1 1 162x x x 3 162x

54x

1 54 1 53x

1 54 1 55x

2) The measures of the angles of a certain triangle are consecutive even integers. Find their measures.

Angle 1 x - 2Angle 2 xAngle 3 x + 2Total 180

2 2 180x x x 3 180x

o60x

o2 60 2 58x o2 60 2 62x

Consecutive Integers

1 2

3

You Try This!

Geometry3) A board is 12 ft long and is to be cut into 3 pieces so that the second piece is twice the size of the first piece, and the third is three times the size of the second piece. Find the length of the 3 pieces of board.

Analysis: second = 2 · first, third = 3 · second = 3 · (2 · first) = 6 · first

1 2 2 3 3 3 3 3 3

Piece 1 xPiece 2 2xPiece 3 6x

2 6 12x x x 9 12x

4 82 2 ft.

3 3x

46 6 8 ft.

3x 4

ft.3

x

Geometry4) The longest side of a triangle is 3 inches more than twice the middle side. The shortest side is 2 inches less than the middle side. If the perimeter is 45 inches, how long is each side?Longest 2x+3Middle xShortest x – 2 Perimeter 45

2 3 2 45x x x 4 1 45x

11 in.x

2 3 2 11 3 25 in.x

4 44x

2 11 2 9 in.x

2x+3

x

x-2

You Try This!

5) A pair of hikers, 18 miles apart, begin at the same time to hike toward each other. If one walks at a rate that is 1 mph faster than the other, and if they meet 2 hours later, how fast is each one walking?

Hiker 1 x 2 2x

Hiker 2 x+1 2 2(x+1)

Rate Time Distance = 18

2 2( 1) 18x x 2 2 2 18x x 4 2 18x

4 16x 4 mphx

1 4 1 5 mphx

Traveling

Hiker 1’s distance + Hiker 2’s distance = 18

Hiker 1’s dist. Hiker 2’s dist.

18

6) A pair of cars, 280 miles apart, begin at the same time to run toward each other. If car A from city A runs at a rate that is 10 mph faster than car B from city B, and if they meet 2 hours later, how far is the place they meet away from city A?

Car A x + 10 2 2(x + 10)

Car B x 2 2x

Rate Time Distance = 280

2 2( 10) 280x x 2 2 20 280x x 4 20 280x

4 260x 65 mphx

10 75 mphx

Traveling

Car A’s distance + Car B’s distance = 280

Car A’s dist. Car B’s dist.

280

A B

You Try This!

2( 10) 2 75 150 mi.x

7) The Yankee Clipper leaves the pier at 9:00am at 8 knots (nautical miles per hour). A half hour later, The Riverboat Rover leaves the same pier in the same direction traveling at 10 knots. At what time will The Riverboat Rover overtake The Yankee Clipper?

Yankee Clipper

9:00 ~ 9:30

Traveled

4 nt. miles

8 x hours after 9:30

8x 8x + 4

Riverboat Rover

9:00 ~ 9:30

Traveled

0 nt. miles

10 x hours after 9:30

10x 0 + 10x

rate time dist. total

4 8 0 10x x 4 2x

2 hr.x

Yankee Total = Riverboat Rover Total

Traveling

4 nt. mi.

RR

YC

RR

YC

10x nt. mi.

8x nt. mi.YC

9:00 9:30 x hr. after9:30

8) The school play sold 450 tickets for a total of $1160. If student tickets are $2.00 and adult tickets are $4.00, how many of each type were sold?

Student 2 x 2xAdult 4 450 – x 4(450 – x) Total ----- 450 1160

2 4(450 ) 1160x x 2 1800 4 1160x x

2 640x

320 tksx

2 1800 1160x 450 450 320 130 tksx

Tickets

Student tickets sales + Adult tickets sales = 1160

9) Fred is selling tickets for his home movies. Tickets for friends are $3.00 and everyone else must pay $5.00 per ticket. If he sold 72 tickets and made $258 how many of each type did he sell?

Friend 3 x 3xNon-Friend 5 72 – x 5(72 – x) Total ---- 72 258

3 5(72 ) 258x x 3 360 5 258x x

2 102x

51 tksx

2 360 258x 72 72 51 21 tksx

TicketsYou Try This!

10) Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?

Barney 450 3 x 450 – 3xBetty 120 8 x 120 + 8x

initial wk spend wk end total

450 3 120 8x x

450 120 11x

3 wkx 330 11x

Accounting

Fred 100 4 x 100 + 4xWilma 28 10 x 28 + 10x

initial wk sp wk end total

100 4 28 10x x

100 28 6x

12 wkx 72 6x

AccountingYou Try This!

28 10 28 10 12 $148x

11) Fred has $100 and saves $4 each week. Wilma has $28 and saves $10 each week. How long will it take for them to have the same amount of money? What is that amount?

More on Consecutive Integers12) Find three consecutive integers that the difference of

the product of two larger ones and the product of two smaller ones is 30.

Integer 1 x - 1Integer 2 xInteger 3 x + 1Prod. of Larger 2 x(x + 1)Prod. of Smaller 2 x(x - 1)

( 1) ( 1) 30x x x x 2 2 30x x x x 2 30x

15x 1 16x 1 14x

More on Traveling

13) A driver averaged 50mph on the highway and 30mph on the side roads. If the trip of 185 miles took a total of 4 hours and 30 minutes, how many miles were on the highway.

Highway 50 x x/50Side Road 30 185 – x (185 – x)/30Total 185 4.5

1854.5

50 30

x x My God! It is so

complicated!!!

Highway 50 x 50xSide Road 30 4.5 – x 30(4.5 – x)Total 4.5 185

50 30(4.5 ) 185x x 50 135 30 185x x 20 135 185x

20 50x 2.5 hr.x

50 50(2.5) 125 mi.x

13) A driver averaged 50mph on the highway and 30mph on the side roads. If the trip of 185 miles took a total of 4 hours and 30 minutes, how many miles were on the highway.

More on Traveling

Weighted Averages14) You have 32 coins made up of dimes and

nickels. You have a total of $2.85. How many of each type of coin do you have?

Dime 10 x 10xNickel 5 32 – x

5(32 – x)

Total 32 28510 5(32 ) 285x x 10 160 5 285x x 5 160 285x

5 125x 25x

32 7x

15) The Quick Mart has two kinds of nuts. Pecans sell for $1.55 per pound and walnuts sell for $1.95 per pound. How many pounds of walnuts must be added to 15 pounds of pecans to make a mixture that sells for $1.75 per pound.

Pecans 1.55 15 15 · 1.55Walnuts 1.95 x 1.95xMixture 1.75 x +15 1.75(x + 15)

1.55 15 1.95 1.75( 15)x x 23.25 1.95 1.75 26.25x x 23.25 0.2 26.25x

0.2 3x 15 lb.x

Weighted Averages

16) A druggist must make 20 oz of a 12% saline solution from his supply of 5% and 15% solutions. How much of each should he use?

12% solution

12% 20 20·12%

5% solution

5% x x · 5%

15% solution

15% 20 – x (20 – x) ·15%

20 0.12 0.05 (20 ) 0.15x x 2.4 0.05 3 0.15x x

0.6 0.1x

6 oz.x 2.4 0.1 3x

Mixture

20 20 6 14 oz.x

3.6 Solving Decimal EquationsObjective: To find exact and approximate solutions of equations that contain decimals.

Solve the equation. Round your result to the nearest hundredth.

1) 26133 x 2) 1697 x

29x1467)3 43x3)11x2(12)4

)x29()x73(14)5 x55.637.982.4x39.2)6

Solve the equation. Round your results to the nearest thousandth.

)x78.29(3.0)x4.721.3(3.1)7

Solve the equation. Round your results to the nearest thousandth.

x108.3x14)1x10(4.2)8

9)Jenny and some of her friends are going to see a movie. Jenny has generously offered to pay for all of the tickets and the snacks. The cost of a student’s ticket is $6.25, and their group spent a total of $48.36 on snacks. If the total amount of money spent is $85.86, how many movie tickets did Jenny buy?

10) Max goes to the deli and buys some ham and some turkey. Ham costs $4.29 per pound and turkey costs $3.89 per pound. If Max purchased 1.5 pounds of ham, and his total bill was $15.19, how much turkey did he buy? Round your answer to the nearest hundredth.

Summary

Solving the decimal equation is the same as solving the any other equation we have learned.

Assignment

P. 169 #’ 14 - 38

3.7 Formulas and Functions Objective:

1. Solve a formula for one of its variable2. Rewrite an equation in function form

Formula -- an algebraic equation that relates two or more real-life quantities.

Example 1 Solving and using an area formulaa) Solve A = lw for w. (In other words, isolate w.)

b) Find the width of a rectangle that has an area of 42 ft2 and a length of 6 ft.

A lw

l l

Aw

l

427

6

Aw

l

Example 2 Solving and using an area formulaa) Solve I = Prt for t.

b) Find the number of years t that $2800 was invested to earn $504 at 4.5%.

Pr

Pr Pr

I t

Pr

It

5044

Pr 2800 (0.045)

It

Example 3 Solve: V = r2h for h

r2 r2

= h

Example 4 Solve: 3x + y = 4 for y

y = –3x + 4

-3x -3x

V = r2h

Vr2

Practice 1 Solve: A = LW for L

A W

W W

Practice 2 Solve: I = P r t for P

r t r t

I rt

P =

= L

A = LW

I = P r t

Example 5 Solve: –3x – 5 = 13

– 3x = 18

+5 +5

– 3x = 18 –3 –3

1 type inv. op.3.2 question

1 type inv. op.

x = 6

Example 6 Solve: mx – a = k for x

mx = k + a

+a +a

mx = k + a m m

1 type inv. op.3.2 question

1 type inv. op.

k + a m

x =

Example 7 Solve for y:

–8x –8x

y = –12x +

3 8xy3

2

3 8x y3

2

3)8x (y3

2

2

3

2

3

2

9

Practice Solve for y

1. 4x + y = 11 2. –3x – 4y = –10

3. 4. 3x 6xy4

35 2x

5

y

Rewriting an equation in Function Form

A two-variable equation is written in function form if one of its variables is isolated on one side of the equation. The isolated variable is the output and is a function of the input. For instance, the equation P = 4s describes the perimeter P of a square as a function of its side length s.

2 9x y

Example 8 Rewrite the equation 2x – y = 9, so that y is a function of x.

-2x -2x

2 9y x

9 2y x -1 -1

Example 9a) Write the equation 2x – y = 9, so that x is a function of y.

b) Use the result to find x when y = –2, –1, 0 and 1

2 9x y + y + y

9

2

yx

2 9x y 2 2

y x

-2 (–2 – 9)/2 = –5.5

-1 (–1 – 9)/2 = –5

0 (0– 9)/2 = –4.5

1 (1– 9)/2 = –4

Example 10 Solve for y: 4(y + 2) = 8 – 3x

4 4

–2 –2 4y = –3x 4 4

4(y + 2) = 8 – 3x

4(y + 2) = 8 – 3x

y + 2 = 4

3x

4

8

y = 4

3x

4y + 8 = 8 – 3x

–8 –8

4y = – 3x

y = 4

3x

Example 11 Solve for F. 5

( 32) 2739

K F

5( 32) 273

9K F

-273 -2735

273 ( 32)9

K F

9273 32

5K F

9 9 5273 32

5 5 9K F

+32 +32

9273 32

5K F

2 2

Example 12 Solve for y:

y = 10x – 3

y + 3 = 10x–3 –3

2 2

7x2x 3) (y 2

1

7x2x 3) (y 2

1

5x 3) (y 2

1

5x 3) (y 2

1

7x2x 3) (y 2

1

5x 3) (y 2

1

5x 2

3 y

2

1

2

3

2

3

2

35x y

2

1

) (2

35x y

2

1

y = 10x – 3

–2x –2x –2x –2x

Example 13 Solve for w:

–12x – 12x

w = –20x+12

32x 4

12xw

w + 12x = –8x+12

3)2x( 4

12xw

4

4

3)2x( 4

12xw

41

4

Practice Solve for w

1. 2. 4 5x)(w7

2 23y

5

4wy

Example 14 The Pathfinder was launched on December 4, 1996. During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.

a. 30,000 miles per hourb. 40,000 miles per hourc. 60,000 miles per hourd. 80,000 miles per hour

We must use the formula d = r t and solve for the time t.

10333.332430.555 days

24

310,000,00010333.332 (h)

30,000

dtr

Example 14 The Pathfinder was launched on December 4, 1996. During its 212-day slight to Mars, it traveled about 310 million miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.

a. 30,000 miles per hourb. 40,000 miles per hourc. 60,000 miles per hourd. 80,000 miles per hour

We must use the formula d = r t and solve for the time t.

7750322.917 days

24

310,000,0007750 (h)

40,000

dtr

Example 14 The Pathfinder was launched on December 4, 1996. During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.

a. 30,000 miles per hourb. 40,000 miles per hourc. 60,000 miles per hourd. 80,000 miles per hour

We must use the formula d = r t and solve for the time t.

5166.667215.278 days

24

310,000,0005166.667 (h)

60,000

dtr

Example 14 The Pathfinder was launched on December 4, 1996. During its 212-day slight to Mars, it traveled about 310 miles. Estimate the time the Pathfinder would have taken to reach Mars traveling at the following speeds.

a. 30,000 miles per hourb. 40,000 miles per hourc. 60,000 miles per hourd. 80,000 miles per hour

We must use the formula d = r t and solve for the time t.

3875161.458 days

24

310,000,0003875 (h)

80,000

dtr

Which of the four speeds is the best estimate of Pathfinder’s average speed?

Summary

1. Solving a formula for one of its variable is the same as solving an equation.

2. Rewriting an equation in function form is the same as solving a formula for one of its variable.