6-1 solving systems by graphing - ktl math...

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Vocabulary

Review

Chapter 6 170

Solving Systems by Graphing6-1

Write I if the amount described is infinite. Write F if the amount is finite.

1. the rational numbers greater than 6 2. the number of seats in a movie theater

3. the number of grams in one kilogram 4. the set of odd numbers

5. Give one example of an infinite amount. Explain why the amount is infinite.

________________________________________________________________________

________________________________________________________________________

Vocabulary Builder

system (noun) SIS tum

Other Word Forms: systematic (adjective), systematize (verb)

Main Idea: A system of linear equations has two or more linear equations. A solution to this system is an ordered pair that makes all of the equations true.

Use Your Vocabulary

Complete each statement with the appropriate word from the list.

systematic system systematize

6. The librarian planned to 9 the donated magazines.

7. The American 9 of government is based on the Constitution.

8. Sam was 9 in his approach to studying for his final exam.

9. A 9 of linear equations might consist of two equations.

I

I

F

F

Answers may vary. Sample: The set of all positive numbers is one example.

It is infinite because there is no end to the set of positive numbers.

systematize

system

systematic

system

HSM11A1MC_0601.indd 170HSM11A1MC_0601.indd 170 3/8/09 1:12:22 PM3/8/09 1:12:22 PM

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d.Problem 1

Problem 2

HSM11_A1MC_0601_T91200

x

y

O 42

2

4

–2–4

–2

–4

HSM11_A1MC_0601_T91201

Write

Relateactivation

feenumber

of monthsmonthlychargetotal cost timesis plus

$10 $20c n? 15

? 15

Let c 5 total costDefine

Service 1:

ncService 2:

.

Let n 5 month number .

$ 15$ 11

171 Lesson 6-1

Solving a System of Equations by Graphing

Got It? What is the solution of the y 5 2x 1 4 system? Use a graph. Check your answer. y 5 x 1 2

10. Graph both lines on the coordinate plane at the right.

11. Circle the point of intersection of the two lines.

(2, 0) (0, 2) (22, 0) (22, 2)

12. Check that your answer to Exercise 11 makes both equations true.

y 5 2x 1 4 y 5 x 1 2

0 2 ? ( ) 1 4 0 1 2

0 1 4 5

5

13. The solution of the system is ( , ).

Writing a System of Equations

Got It? One satellite radio service charges $10 per month plus an activation fee of $20. A second service charges $11 per month plus an activation fee of $15. In what month was the cost of the service the same?

14. Complete the model below.

15. Which variable will you graph on the horizontal axis?

Which variable will you graph on the vertical axis?

16. What will the intersection of the graphs of the two linear equations tell you?

_______________________________________________________________________

_______________________________________________________________________

Answers may vary. Sample: The intersection will show when (in

what month) the cost of the services is the same.

0 0

0 0 0

0

n

c

0

0

22

24

22

22

FPL ProofAlgebra 1 On Level CompanionHSM11A1MC_0601.indd 171 6/15/10 7:42:58 AM

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d.Problem 3

n

c

O 98761 2 3 4 5

100

10

2030

40

50

60

70

80

90

Tota

l Cos

t ($)

Number of Months

x

y

O 642

42

6

46

246

y

O 642

42

6

46

246x

y

O 642

42

6

46

246x

Chapter 6 172

17. Graph the equations you wrote in Exercise 14 on the coordinate grid at the right.

18. Multiple Choice Which ordered pair gives the coordinates of the point of intersection of the two lines?

(0, 0) (5, 70)

(1, 30) (70, 5)

19. So, in month the total cost of

either service is the same.

A system of equations that has at least one solution is consistent. A consistent system can be either independent or dependent.

A consistent system that is independent has exactly one solution. A consistent system that is dependent has infi nitely many solutions.

A system of equations that has no solution is inconsistent.

Underline the correct word, words, or number to complete each sentence.

20. If two lines intersect at one point (have different slopes), the system of

equations is independent and consistent / inconsistent . The system of equations

has 0 / 1 / infinitely many solution(s).

21. If two lines are the same (have the same slope and y-intercept), the system of

equations is dependent and consistent / inconsistent . The system of equations has

0 / 1 / infinitely many solution(s).

22. If two lines are parallel (have the same slope and different y-intercepts),

the system of equations is consistent / inconsistent . The system of equations has

0 / 1 / infinitely many solution(s).

Systems With Infinitely Many Solutions or No Solution

Got It? What is the solution of the system? Use a graph. y 5 2x 2 3Describe the number of solutions. y 5 2x 1 5

23. Multiple Choice Which shows the solution of the two equations?

24. The lines are parallel / perpendicular . The system has no solution / one solution .

5


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Math Success

Lesson Check

Lesson Check

y

O 84 1612

4

8

12

16

481216

4

8

12

16

x

173 Lesson 6-1

Check off the vocabulary words that you understand.

system of linear equations solution of a system of linear equations

consistent independent dependent inconsistent

Rate how well you can solve a system of linear equations.

Solve the system by graphing.

y 5 12x 1 6

y 5 x 2 2

25. The slope of y 5 12x 1 6 is .

The y-intercept of y 5 12x 1 6 is .

26. The slope of y 5 x 2 2 is .

The y-intercept of y 5 x 2 2 is .

27. Graph each line in the system on the coordinate grid at the right.

28. The solution of the system

is ( , ).

Vocabulary Draw a line from each type of system in Column A to the number of solutions the system has in Column B.

Column A Column B

29. inconsistent exactly one solution

30. consistent and dependent infinitely many solutions

31. consistent and independent no solution

• Do you know HOW?

• Do you UNDERSTAND?

12

6

16 14

22

1


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Vocabulary

Review

Chapter 6 174

6-2 Solving Systems Using Substitution

1. Cross out the expression that does NOT include a variable.

y 1 9 a 2 b 23 1 9 3x 1 4y 1 12

2. Circle the equation in which the variable is isolated.

8k 5 16 m 1 3 5 22 a 5 7 2 3 12 5 z 1 4

Vocabulary Builder

substitution (noun) sub stuh TOO shun

Related Words: substitute (verb or adjective)

Definition: A substitution is something taking the place of something else.

Example: A substitution of 4 for x and 8 for y in x 1 y gives 4 1 8, or 12.

Use Your Vocabulary

Complete each statement with the appropriate form of the word substitution.

3. ADJECTIVE We had a 9 teacher in social studies class today.

4. NOUN The coach made a 9 of one player for another.

5. VERB To evaluate the expression x 1 6, you can 9 a number for x.

6. Write a combination of coins that you could substitute for each dollar amount.

$1.00 $2.00 $5.00

You can solve linear systems by solving one of the equations for one of the variables. Th en substitute the expression for the variable into the other equation. Th is is called the substitution method.

substitute

substitution

substitute

Answers may vary. Sample: 10 dimes

Answers may vary. Sample: 40 nickels

Answers may vary. Sample: 20 quarters


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d.Problem 1

I need to solve one of the equations forone of the variables. Solving the secondequation for x is quickest.

Next I substitute into the otherequation. Then I solve the equationfor y.

Now I substitute the y-value into eitheroriginal equation and solve for x.

x 1 3y 5 27x 5 27 2 3y

6y 1 5x 5 8

6y 1 5( ) 5 87 3y

6y 1 y 5 8

5

35 15y

9y 43

<y 4.8

x 1 3y 5 27

x 1 3( )5 27

x < 7.4

4.8

Think Write

Problem 2

175 Lesson 6-2

Using Substitution

Got It? What is the solution of the system? Use substitution. y 5 2x 1 7Check your answer. y 5 x 2 1

7. Circle the equation that shows a substitution from one equation into the other.

2x 1 7 5 x 2 1 y 5 x 2 1 y 5 2x 1 7 2y 1 7 5 y 2 1

8. Now find the value of x. 9. Use the value of x to find the value of y.

10. The solution is ( , ).

11. Check your answer by substituting the values for x and y in both equations.

y 5 2x 1 7 y 5 x 2 1

0 2 ? ( ) 1 7 0 2 1

0 1 7 5

5

Solving for a Variable and Using Substitution

Got It? What is the solution of the system? Use substitution. 6y 1 5x 5 8 x 1 3y 5 27

12. Complete the reasoning model below.

13. The solution is about ( , ).

2x 1 7 5 x 2 1x 1 7 5 21

x 5 2 8

y 5 2x 1 7y 5 2 ? (28) 1 7y 5 29

28 29

29

29

29 29

28

216

29 28

29 29

24.87.4


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Problem 3

Relate

Write

total games number of $2 gamesnumber of $4 gamesis plus

y15 x6

22 x?4

Relate

Write

total cost cost of $2 gamescost of $4 gamesis plus

2 ? y15

Chapter 6 176

Using Systems of Equations

Got It? You pay $22 to rent 6 video games. The store charges $4 for new games and $2 for older games. How many new games did you rent?

14. Define the variables.

Let x 5 the number of $4 games.

Let y 5

.

15. Complete the models below.

16. Solve the first equation for y.

17. Substitute your answer from Exercise 16 to find the x-value.

18. The student rented new ($4) games.

If you get an identity, such as 2 5 2, when you solve a system of equations, then the system has infi nitely many solutions. If you get a false statement, such as 8 5 2, then the system has no solution.

the number of $2 games

6 5 x 1 y6 2 x 5 y

22 5 4x 1 2y22 5 4x 1 2(6 2 x)22 5 4x 1 12 2 2x10 5 2x

5 5 x

5


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d.Problem 4

Now Iget it!

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0 2 4 6 8 10

Math Success

Lesson Check


substitution system of equations solution of a system

Rate how well you can solve systems using substitution.


solved for x:

x 5 12 2

12y

solved for y: y 5 2x 2 1

solved for x:

x 5 3 2 23y

solved for y: y 5 6 2 2x

177 Lesson 6-2

Systems with Infinitely Many Solutions or No Solution

Got It? How many solutions does the system have? 6y 1 5x 5 8 2.5x 1 3y 5 4

19. Use substitution to solve the system of equations.

20. I obtained an identity / a false statement , so this system of equations has

infinitely many / no solutions .

For the system, tell which equation you would first use to solve for a variable in 22x 1 y 5 21the first step of the substitution method. Explain your choice. 4x 1 2y 5 12

21. Each of the equations has been solved for a variable. Explain which variable you would choose to solve for and why.

Equation 1 Equation 2

______________________________________________________________________

______________________________________________________________________

2.5x 1 3y 5 4 2.5x 5 23y 1 4 x 5 21.2y 1 1.6

6y 1 5(21.2y 1 1.6) 5 8 6y 2 6y 1 8 5 8 8 5 8

Check students' work. Sample:

Answers may vary. Sample: Solving the first equation for y is easy because

its coefficient is 1.


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Vocabulary

Chapter 6 178

6-3 Solving Systems Using Elimination

Review

1. Multiple Choice Which of the following does NOT describe a formula?

a rule or principle a fi xed method for doing something

a recipe or prescription a constant

2. Circle the formula you can use to find the volume of a rectangular prism.

2(/ 1 w) πr2 /hw s2

3. Cross out the pairs of equations that are NOT equivalent.

3y 1 2x 5 24; y 5 23x 1 8 4a 1 b 5 17; b 5 17 2 4a

2m 1 6n 5 10; m 1 3n 5 5 j 1 k 5 11; j 5 k 1 11

Vocabulary Builder

elimination (noun) ee LIM in ay shun

Other Word Forms: eliminate (verb), eliminated (verb), eliminating (verb)

Definition: Elimination is the act of removing something.

Example: A gardener works toward the elimination of weeds from a garden.

Math Usage: In elimination, you use properties of equality to add or subtract equations to eliminate a variable in a system.

Use Your Vocabulary

Complete each sentence with the word elimination or one of its other word forms.

4. The city council will not 9 funding for the public library.

5. The 9 of the dance portion of the contest was disappointing.

6. The new floor cleaner is great for 9 dirt.

eliminate

elimination

eliminating


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d.Problem 1

Problem 2

179 Lesson 6-3

Solving a System by Adding Equations

Got It? What is the solution of the system? Use elimination. 5x 2 6y 5 2323x 1 6y 5 48

7. Reasoning Why should your first step be to eliminate y?

Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect.

I can’t eliminate x.

If I eliminate x, I can’t add the expressions that contain y.

The expressions 6y and 26y are additive inverses.

8. Circle the equation you get after you eliminate y.

22x 5 216 2x 5 16 8x 5 16 28x 5 16

9. Solve the equation for x. Use that value to solve for y.

Solve the equation for x. Use the x-value to solve for y.

? x 5

? x 5

x 5

10. The solution of the system is ( , ).

Solving a System by Subtracting Equations

Got It? Washing 2 cars and 3 trucks takes 130 min. Washing 2 cars and 5 trucks takes 190 min. How long does it take to wash each type of vehicle?

11. Use the equations at the right for this system. 2c 1 3t 5 1302c 1 5t 5 190

Let c 5

.

Let t 5

.

12. Eliminate the variable c. Then solve for t.

2c 1 3t 5 130 2c 1 3t 5 130

2c 1 5t 5 190 22c 1 ? t 5 2190

0 1 ? t 5 260

t 5

To subtract the second row from the fi rst, change to opposite signs and add down.

3x 1 6y 5 48

3 ? 1 6y 5 48

6y 5 48 2

6y6 5

6

y 5

time to wash a car

time to wash a truck

8

8 8

2

8 16

16

✗

✗

✓

2

6

42

2

7

7

22

25

30


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Problem 4

Problem 5

Chapter 6 180

13. Use the value of t to solve one of the original equations for c.

14. It takes min to wash a truck and min to wash a car.

Solving a System by Multiplying Both Equations

Got It? How can you use the Multiplication Property of Equality to change the equations in this system in order to solve it using elimination?

15. How can you eliminate the same variable from both equations?

_______________________________________________________________________

_______________________________________________________________________

16. Write and solve a revised system.

4x 1 3y 5 219

3x 2 2y 5 210

Subtract the two equations.

Solve for the variable.

Substitute the variable and find the solution.

The solution is { }

17. Show that the solution of the revised system is a solution of the original system.

Substitute the solution into each equation.

Finding the Number of Solutions

Got It? How many solutions does the system have? 22x 1 5y 5 7 22x 1 5y 5 12

18. Solve the system.

Multiply by [ ]

Multiply by [ ]

3x 2 2y 5 210 4x 1 3y 5 219

4x 1 3y 5 2193x 2 2y 5 210

Explanations may vary. Multiply the first equation by 3 and the second

equation by 4. Then subtract the equations.

12x 1 9y 5 257

12x 2 8y 5 240

0x 1 17y 5 217

y 5 21

x 5 24

2c 1 3(30) 5 130 2c 5 130 2 90 2c 5 40 c 5 20

22x 1 5y 5 722x 1 5y 5 12

0 5 25

30 20

3

4

f24g, f21g

Sample:

3(24) 2 2(21) 5 210

212 1 2 5 210

210 5 210

4(24) 1 3(21) 5 219

216 1 23 5 219

219 5 219

HSM12A1MC_0603.indd 180 4/28/11 10:42:21 AM

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Lesson Check

181 Lesson 6-3



elimination method solution of a system of linear equations

Rate how well you can find the solution of a system of linear equations by elimination.

Vocabulary If you add two equations in two variables and the sum is an equation in one variable, what method are you using to solve the system? Explain.

20. Add each system of two linear equations together.

A 5x 1 6y 5 30 B 3x 1 7y 5 21 22x 2 6y 5 212 27x 1 2y 5 214

21. Which system of equations, A or B, has a sum that is an equation in one variable?

_______________________________________________________________________

22. How can you use the equation in one variable to solve the system of equations?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

23. When you use an equation in one variable to solve a system of equations, what method are you using to solve the system? Explain.

_______________________________________________________________________

_______________________________________________________________________

19. How many solutions does the system have? Explain.

_______________________________________________________________________

_______________________________________________________________________

Answers may vary. Sample: I am using elimination because one

variable is eliminated when I add the two equations.

System A has a sum that is an equation in one variable.

Answers may vary. Sample: I can find a value for that variable, and then

substitute that value into one of the original equations to find the value

of the other variable.

The system has no solution. Explanations may vary. Sample: When

I solved the system, I got 0 5 25, which is a false statement.

3x 5 18 24x 1 9y 5 7

HSM12A1MC_0603.indd 181 3/19/11 1:08:39 PM

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Vocabulary

Review

NE 23

rd St

.

NE 22n

d A

ve

NE 171st St.

NE 167th St

Bisc

ayne

Blv

d.

W. D

ixie

Hw

y

HSM11_A1MC_0604_T91210

HSM11_A1MC_0604_T91211

x

y

O 2

2

–2

HSM11_A1MC_0604_T91212

x

y

O 2

2

–2

HSM11_A1MC_0604_T91213

x

y

O 42

2

–2

Chapter 6 182

Applications of Linear Systems6-4

1. Multiple Choice Which equation shows what happens when you 3x 2 4y 5 214 use substitution to solve this system of equations? y 5 2x 2 1

2x 2 1 2 4y 5 214 3x 2 (2x 2 1) 5 214

3(2x 2 1) 2 4y 5 214 3x 2 4(2x 2 1) 5 214

Vocabulary Builder

intersection (noun) in tur sek shun

Definition: An intersection is where two or more lines or roads meet.

Math Usage: For two lines, the point of intersection is a point in common where they intersect, or meet.

Use Your Vocabulary

2. The Ancient Spanish Monastery (shown by the star on map) in North Miami Beach, Florida, is thought to be the oldest building in the western hemisphere. Name two streets that intersect at the Ancient Spanish Monastery.

____________________________________________

Write the ordered pair for the point of intersection. If there is no point of intersection, write none.

3. 4. 5.

HSM11_A1MC_0604_T91209

The intersection ofx 1 y = 3 and

x 2 y = 1 is (2, 1).

Sample: NE 167th St. and W. Dixie Hwy.

(0, 21) none (3, 22)

HSM12A1MC_0604.indd 182 3/2/11 10:32:15 AM

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d.Problem 1

Problem 2

HSM11_A1MC_0604_T91214

Write

Relatenumberof books

soldincomeexpenses $2$864 $.80

y 864 1 0.8x 2x

?

numberof books

sold?1

= y =

==

Let x = the number of books sold.Define

Let y = the number of dollars of expense or income .

183 Lesson 6-4

Finding a Break-Even Point

Got It? A puzzle expert wrote a new sudoku puzzle book. His initial costs are $864. Binding and packaging each book costs $.80. The price of the book is $2. How many copies must be sold to break even?


7. Use substitution to solve the system of equations.

y 5 Start with the equation for expenses.

5 Substitute the income expression for y.

5 Subtract 0.8x from each side.

x 5 Divide each side by 1.2.

8. The puzzle expert must sell books to break even.

Identifying Constraints and Viable Solutions

Got It? The zoo has two water tanks that are leaking. One tank contains 10 gal of water and is leaking at at a constant rate of 2 gal/h. The second tank contains 6 gal of water and is leaking at a constant rate of 4 gal/h. When will the tanks have the same amount of water? Explain.

9. How much water would leak from Tank 1 in 2 hours? in 3 hours? How are you getting your answer?

_______________________________________________________________________

_______________________________________________________________________

What algebraic expression could represent how much water is lost from Tank 1 in x hours?

What algebraic expression could represent how much water is lost from Tank 2 in x hours?

10. If there are 10 gallons in Tank 1, how much will be left in 2 hours? in 3

hours? How are you getting your answer?

_______________________________________________________________________

_______________________________________________________________________

864 1 0.8x

864 1 0.8x2x

1.2x 864

720

720

4 gal

4 gal

6 gal

6 gal

2x

4x

Explanations may vary. Sample: The rate of the leak is 2 gal/hr, so multiply

the rate by the number of hours.

Explanations may vary. Sample: Subtract the amount leaked in 2 or 3 hours

from the total number of gallons.

HSM12A1MC_0604.indd 183 4/12/11 2:54:35 PM

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d.

Problem 3

HSM11_A1MC_0604_T91215

Write

Relate

a 1 c

speedof boatin stillwater

speedof thecurrent

downstreamspeed1

5 5 a 2 c 5 2

5

speedof boatin stillwater

speedof thecurrent

upstreamspeed2 5

Let a 5 speed of the boat in still water.Define

Let c 5 the speed of the current .

Chapter 6 184

Solving a Wind or Current Problem

Got It? You row upstream at a speed of 2 mi/h. You travel the same distance downstream at a speed of 5 mi/h. What would be your rowing speed in still water? What is the speed of the current?


15. Now solve the system of equations.

16. Your rowing speed in still water is mi/h.

The speed of the current is mi/h.

11. If y represents the number of gallons of water left in the tanks after x hours, write two equations that represent how much water is left in Tank 1 and Tank 2 after x hours.

Tank 1: y 5 Tank 2: y 5

12. Solve the equations for x and y by substituting the expression for y from the first equation for y in the second equation; then find x and substitute the value of x to find y.

13. Does the answer make sense? Explain.

____________________________________________________________________________

Solutions may vary. Sample:a 1 c 5 5 a 2 c 5 2 2a 5 7 a 5 3.5

Substitute 3.5 for a in a 1 c 5 5.3.5 1 c 5 5c 5 1.5

3.5

1.5

Explanations may vary. Sample: No, because it is not possible to have 22 hours.

10 2 2x 5 6 2 4x 10 1 2x 5 6 2x 5 24 x 5 22

y 5 10 2 2(22) y 5 14

10 2 2x 6 2 4x

HSM12A1MC_0604.indd 184 4/12/11 3:08:48 PM

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Math Success

Lesson Check

185 Lesson 6-4


intersection system of linear equations

substitution elimination

Rate how well you can apply systems of equations.


Reasoning Which method would you use to solve the following system? Explain.

3x 1 2y 5 9 22x 1 3y 5 5

17. Draw a line from each statement in Column A to the most appropriate method for solving a system of equations in Column B.

Column A Column B

One equation is already solved for one of Substitution

Elimination

Graphing

the variables.

A visual display of the equations is needed.

The coefficients of one variable are able to be made the same or opposite.

18. Circle the method you would use to solve the given system.

Graphing Elimination Substitution

19. Explain why you chose that method.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

Explanations may vary. Sample: The equations have coefficients of x that

could be opposites. So, I would multiply the first equation by 2 and the

second equation by 3 to eliminate x.

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Vocabulary

Chapter 6 186

6-5 Linear Inequalities

Review

A solution is a value or values which, when substituted for a variable, make an equation true. Determine the solution of each inequality below. Circle your answer.

1. x 1 17 , 37 2. 2m 1 3 . 15

x . 20 x , 20 m . 6 m , 6

3. Multiple Choice How many solutions does this system of equations have?

0 1 2 infinitely many

Vocabulary Builder

linear inequality (noun) LIN ee ur in ee KWAL uh tee

Definition: Replacing the equals sign in a linear equation with an inequality symbol makes a linear inequality in two variables.

Main Idea: A linear inequality in two variables has an infinite number of solutions, each an ordered pair that makes the inequality true.

Example: y $ 2x 2 9 is a linear inequality in two variables.

Use Your Vocabulary

For Exercises 4–7, write an inequality using the symbol and the equation.

4. . 6x 5 18

5. , m2 5 55

6. $ 3y 5 x 2 26

7. # y 5 2x 1 1

8. Circle the linear inequalities in two variables.

6x . 3 y # 2x 1 5 2x 1 y 5 6 y 5 27 2y . x

n n

linear inequalityy $ 2x 2 9

x 2 5y 5 12x 1 3y 5 9

6x S 18

m2 R 55

3y L x 2 26

y K 2x 1 1


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d.Problem 1

Problem 2

x

y

O

2

24

2

4

4

42

187 Lesson 6-5

Identifying Solutions of a Linear Inequality

Got It? Is (3, 6) a solution of y K 23x 1 4?

9. Complete the steps to determine whether (3, 6) is a solution.

y # 23 ? x 1 4 Write the inequality.

� 23 ? 1 4 Substitute.

� 1 4 Multiply.

� 6 Simplify.

10. Is (3, 6) a solution? Yes / No

For each inequality below, determine whether (3, 6) is a solution.

11. y . x 1 7 12. y # 3x 2 2

Yes / No Yes / No

Graphing an Inequality in Two Variables

Got It? What is the graph of y K 12x 1 1?

Underline the correct word to complete each sentence.

13. The symbol # means greater / less than or equal to.

14. The boundary line will be dashed / solid .

15. Graph the boundary line on the grid.

16. The point (0, 0) is not on the line. Check to see whether that point is a solution of the inequality.

17. Circle the true statement. Then shade the graph.

The side of the line that contains The side of the line that does NOT (0, 0) should be shaded. contain (0, 0) should be shaded.

6 3

26

6

6 S 3 1 76 S 10 False

0 � 12 ? 0 1 10 K 1 ✓

6 K 3 ? 3 2 26 K 9 2 26 K 7 True


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Problem 4

Problem 3

x

y

O 761 2 3 4 5

7

6

1

2

3

4

5

Cash

ews

(lb)

Peanuts (lb)

Chapter 6 188

Graphing a Linear Inequality in One Variable

Got It? What is the graph of the inequality x R 25? Of y K 2?

18. Label each graph.

O 2

2

4

242

4

x

y

O 42

2

4

242

4

x

y

O 42

2

4

42

4

x

y

2 O 42

2

4

242

4

x

y

type of boundary line

equation of boundary line

inequality

19. Describe the graph of each inequality. Use the words boundary line, dashed or solid, and shade or shaded in each description.

x , 25

y # 2

Rewriting to Graph an Inequality

Got It? For a party, you can spend no more than $12 on nuts. Peanuts cost $2/lb. Cashews cost $4/lb. What are three possible combinations of peanuts and cashews you can buy?

20. Let x 5 the number of pounds of peanuts and let y 5

.

21. Complete the steps to solve the inequality for y.

2x 1 4y # 12 Write the inequality.

2x 2 1 4y # 1 12 Subtract the same amount from both sides.

4y#22x

1 Divide each side by the same amount.

y # 1 Simplify.

22. Graph the inequality on the grid.

23. Write three possible combinations of peanuts and cashews that you can buy.

_________________________________________________

_________________________________________________

_________________________________________________

dashed

x 5 4

x S 4

solid

y 5 3

y L 3

dashed

x 5 22

x R 22

solid

y 5 5

y K 5

The boundary line x 5 25 is dashed; the shaded side does not contain (0, 0).

the number of pounds of cashews

Sample: 4 lb of peanuts and 1 lb of cashews,

1 lb of peanuts and 1 lb of cashews, or

0 lb of peanuts and 3 lb of cashews

The boundary line y 5 2 will be solid and the shaded side will contain (0, 0).

2x 22x

12

444

212x 3

Answers may vary.


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Lesson Check

Problem 5

189 Lesson 6-5


linear inequality solution of an inequality

Rate how well you can graph a linear inequality in two variables.


Writing To graph the inequality y R 32x 1 3, do you shade above or below the

boundary line? Explain.

27. Describe how to determine which side of the boundary line should be shaded.

__________________________________________________________________________________

__________________________________________________________________________________

28. Do you shade above or below the boundary line for y , 32x 1 3? Explain.

__________________________________________________________________________________

__________________________________________________________________________________

Writing an Inequality From a Graph

Got It? You are writing an inequality from a graph. The boundary line is dashed and has slope 13 and y-intercept 22. The area above the line is shaded. What inequality should you write?

24. Write the equation of the boundary line in slope-intercept form.

25. Circle the symbol you will use in the inequality.

, # . $

26. Now write the complete inequality.

When a linear inequality is solved for y, the direction of the inequality symbol determines which side of the boundary line to shade. If the symbol is , or #, shade below the boundary line. If the symbol is . or $, shade above it.

Test a point that is not on the line, such as (0, 0). If (0, 0) is a solution, then

so are all the points on the same side of the boundary line as (0, 0).

I would shade below because (0, 0) is a solution and (0, 0)

is below the boundary line.

y 5 13x 2 2

y S 13x 2 2

Answers may vary. Samples are given.

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Review

Chapter 6 190

6-6 Systems of Linear Inequalities

1. Circle the system of equations.

3x 1 2y 5 20 3x 2 2y 5 12 3x 1 2y 5 12 7x 1 2y , 10 4x 1 4y 5 20

2. How can you tell if an ordered pair is a solution to a system of equations?

_______________________________________________________________________

_______________________________________________________________________

Vocabulary Builder

boundary (noun) bown dree

Related Words: bounds (noun or verb), border (noun or verb), bounded (adjective)

Definition: A boundary is something that divides one item from another.

Math Usage: The graph of a linear inequality in two variables is a region bounded by a line. All points on one side of the boundary line are solutions.

Use Your Vocabulary

3. Complete each sentence with bounds, boundary, or border.

The landscaper planted a 9 of flowers along the sidewalk.

The ball bounced out of 9.

The 9 line on the map between countries is shown.

4. Circle the ordered pair that is on the boundary of y 1 2 # x 2 5.

(9, 0) (0, 26) (6, 21) (10, 5) (5, 23)

HSM11_A1MC_0606_T91224

O

x

y

boundary line

Answers may vary. Sample: An ordered pair is a solution to

a system of equations if it makes all of the equations true.

border

bounds

boundary

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d.Problem 1

Problem 2

HSM11_A1MC_0606_T91225

x

y

O

4

–4–8

–4

8

84

–8

x

y

O

2

�2 2

191 Lesson 6-6

Graphing a System of Inequalities

Got It? What is the graph of the system? y L 2x 1 5

23x 1 y K 24

5. The boundary line for y $ 2x 1 5 will be solid / dashed .

The boundary line for 23x 1 y # 24 will be solid / dashed .

6. Write the inequality 23x 1 y # 24 in slope-intercept form.

y # ? x 2 4

7. Graph the inequalities on the coordinate grid at the right.

Writing a System of Inequalities From a Graph

Got It? What system of inequalities is represented

by the graph?

8. Find the inequality shown by the dashed line. 9. Find the inequality shown by the solid line.

Determine the equation of the dashed line. Determine the equation of the solid line.

y 5 y 5

The shaded region is above / below the The shaded region is above / below the

line. The boundary line is solid / dashed . line. The boundary line is solid / dashed .

Circle the inequality symbol you should use Circle the inequality symbol you should use to write the inequality for this graph. to write the inequality for this graph.

. , # $ . , # $

10. Write the system of inequalities represented by the graph.

3

212x 1 1 1

2x 1 1

y , 212 x 1 1

y # 12 x 1 1

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Problem 3

Define what each variable represents.

Let x 5 the width of the dog run.

Let y 5 the length of the dog run .

Graph the system of inequalities.

Write a system of inequalities.

2x 1 2y

x

y

126 2x 1 63

10

50

1

2

3

In slope-intercept form, this is y .

y

O 9080706010 20 30 40 50

70

60

10

80

90

20

30

40

50

Leng

th (f

t)

Width (ft)

x

Chapter 6 192

Using a System of Inequalities

Got It? You want to build a fence for a rectangular dog run. You want the run to be at least 10 ft wide. The run can be at most 50 ft long. You have 126 ft of fencing. What is a graph showing the possible dimensions of the dog run?

Remember, a dog run must have fencing on all four sides.

11. Follow the steps to write a system of inequalities and graph it.

12. Why is the graph only in the first quadrant?

_______________________________________________________________________

_______________________________________________________________________

13. Use the graph to write possible dimensions of the dog run.

width (x): width (x): width (x):

length (y): length (y): length (y):

Answers may vary. Sample: The graph is only in the first quadrant because

the values are about length, and length cannot be negative.

20 feet

43 feet

30 feet

30 feet

25 feet

38 feet

Answers may vary. Samples given.

HSM11A1MC_0606.indd 192HSM11A1MC_0606.indd 192 4/14/09 10:46:39 AM4/14/09 10:46:39 AM

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Lesson Check

Math Success

x

y

x

y

x

y

193 Lesson 6-6


system of linear inequalities solution of a system of linear inequalities

Rate how well you can solve a system of linear inequalities.


Reasoning Suppose you are graphing a system of two linear inequalities, and the boundary lines for the inequalities are parallel. Does that mean that the system has no solution? Explain.

14. Draw a line from each graph to the corresponding system of linear inequalities.

y # 234x 1 6 y $ 2

34x 1 6 y $ 2

34x 1 6

y $ 234x 1 3 y # 2

34x 1 3 y $ 2

34x 1 3

15. Which of the systems have solutions?

__________________________________________________________________________

__________________________________________________________________________

16. Would the system of y # 234x 1 6 and y # 2

34x 1 3 have a solution? If so,

describe the solution.

__________________________________________________________________________

17. Is it possible for a system of two linear inequalities with parallel boundary lines to have a solution? Yes / No

18. Is it possible for a system of two linear inequalities with parallel boundary lines to have no solution? Yes / No

The systems shown by the second graph and the third graph have solutions.

The system shown by the first graph does not.

Yes. Answers may vary. Sample: The solution would be y K 234x 1 3.


6-1 solving systems by graphing - ktl math...

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