05x -L- 6y = 0 2x + y = 7

516) + 6(-5) = 0 2(6) + (-5) = 7 30 - 30 = 0 1 2 - 5 = 7

0 = 0 7 = 7 0 y = - 2 y = 4x

-8 = -2 False -8 = 4(-2)

Ox + y = 5 y = -x + 5

--8 = -8 x - y = 1 -y = -x + 1

y = x - 1

0 x - y = -2 -y = -x - 2 y = x + 2

4x - y = 4

-y= -4x + 4 y = 4x - 4

0 4x + y = 7 y = -4x + 7

x - 3y = 18 -3y = -x + 18 y =x - 6

3.

:WWW.F7•, 01n4 .-;

x + 2y= 10

2y= -x 10 -1

= x 5

9x + 2y = -6 2y = -9x - 6

y = 2 x - 3

= 2 61•

• A. Exercises

Is (-2, 5) a solution to each system?

24. 6-2 • 2-i 3 ' 31 3 9- 3 -

527 1 25. 144 + 72 -

3. x - y = 7 yeS 2x + 3y = 4 0 0

4. 3x + 2y = 4

n 5x + 3y = 5 €. S

system determine whether the given point is the solution.

1.x + y = 3

2x + y = 1

2.x 2y = 8

3x + y -= 1

For each

5. x + 3y = 13; (2, -5)

8. 3x - y =- -7, (-1, 4)

6> + y 7 el x - 5y --= -19 na

6. 2> - 5y = 9, (7, 1)

9. 5x + 6y = 0; (6, -5)

x + y = 8 vfotS 2x + y = 7 tcs

7. x + 2y = 1; 3. -2)

10. y = -2, (-2, -8)

3> - 2y = 5 rNO y -= 4x

B. Exercises Graph each system of linear equations to find its solution.

11. x y = 5

16. 4x + y = 7 x - y = 1 '3 2' x - 3y 18 13 . -

12. x - y = -2 17. x + y = 9

- y 4

11x + y 19

13. 6> + y = 7 18. 9x - y = -5 y = 1

12x - y -8 -4)

14. x + 2y = 10

19. x - 5y = -30

9y + 2y= -6

2x - y = 3

15. 8i - 3y = -12

- y - 8 -3, -4)

C. Exercises Graph the following equations and estimate the solution of the system.

20. 9x - 2y = 7

3x + 4y 14 student answers should approximate the solution (1-,.

Currnilati.ve Review Simplify.

3 12 49 37 ,; 21. 5 + 35 • 9 15 12.31

22. I 2614 1 -

r 6 39 -3 ,'*-1

1 8 - • N 343 7

7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING 275

8 8x - 3y = -12 4x - y = -8 -3y = -8x - 12 -y = -4x - 8

8 y = Tx + 4 y = 4x + 8

ii

r I ! cot-

!

7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING 275

gx-Fy.-- y =--x+ 4

nnsistent; independent

y

5x + y = 8 — 5x + 8

Notice that inconsistent means not consistent and independent means not dependent.

POSSIBLE SOLUTIONS FOR SYSTEMS OF EQUATIONS

Consistent Inconsistent

Independent Dependent

Number of solutions finite infinite none

Graphs graphs graphs graphs do not intersect coincide intersect

Applied to lines (number of solutions)

one all points on the line

none

► A. Exercises

Solve each system by graphing and tell whether it is consistent or inconsistent. If the system is consistent, tell whether it is dependent or independent.

graph marked off by 2's

® + y = 8 y —2x + 8

4x + 2y = 16 2y= —4x + 16 y = —2x + 8

consistent; c ependent

y

Case 3 1. Lines coincide. 2. Solution is an infinite set. 3. System is consistent and dependent.

Example x + y = 3 2x + 2y = 6

01► 3x -y= 5 y —3x + 5

4x + 3y = 15 3y = —4x + 15

— y= 34 x + 5

Since the slope-intercept form of these two equations is the same, their graphs are identical. The line described by the equation y = —x -1- 3 is the graph for both equations. Thus. the entire line is the solution to the s y stem. The line contains an infinite number of points all of which are solutions.

Any linear system of equations that produces an infinite number of solutions

is called consistent but dependent (not independent).

To summarize, a system of two linear equations may have 0. 1, or an infinite number of solutions. Use the slope-intercept form of the equations to deter-mine the number of solutions.

Definitions

A consistent system of equations is a system that has at least one solution.

A dependent system of equations is a consistent system that has an infinite number of solutions.

Ox — y = 3 —y =—x+ 3 y = x — 3

4x — y = 15 —y= —4x + 15 y = 4x — 15

1. x + y = 4 3); consistent: 5x + y = 8 independent

2. x — y = 3 (4. 1); consistent, 4x — y = 15 independent

3. 2x + y = 8 :,ntire line; consistent, 4x + 2y = 16 dependent

4. 3x + y = 5 i".;; 5); consistent,

4x + 3y = 15 independent

5. x + 2y — 10 = 0 4); consisten',.

3x + 2y = 14 independent

6. 8x — 2y = 6 no solution;

y 4x + 3 inconsistent

7. x — 2y = 4 1—.7,, —3); consistent,

3x + 2y = —12 independent

8. x + 2y = 8 ern:Pie line; consistent,

y = —1+ 4 dependent

278 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

As you discuss this case, point out that the

slopes and y-intercepts of the two equations

are the same and that the equations are equiv-alent. Ultimately, the equations for a depend-ent system are the same and the solution is the entire line. This information is summarized for the students in the table on page 278. (In nonlinear dependent systems. the solutions of one equation are a subset of another.)

Have the students solve the systems in examples 1, 2, and 3 by graphing the equations and, identify the type of system.

Common Student Error. Students may have difficulty with the terminology in this section. Speak carefully and accurately as a model for them. n

278 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES

8x - 2y = 6 -2y = -8x + 6 y = 4x - 3

y = 4x + 3

X

x 2y = 3 2y = -x -- 8

y= 21 x + 4

y= 21 x + 4 (tx = —3

incons'stent

3 x_ y = 6 -y = -x + 6 y = x - 6

5x — 5y = 30 —5y = —5x + 30 y = x - 6

x = 4

y

7.2 USING THE GRAPHING METHOD 279

$•• B. Exerciies

Solve each system by graphing and tell whether it is consistent or inconsistent. If the system is consistent, tell whether it is dependent or independent.

9. x= —3 15. 3x + 4y = 4 -x= 4 .'7" x — 2y = 8 ..ant

10. x — y = 6 _ 16. 3x + y = 5( ;iista t,

— 5y = 30 .dependent y = 3x — 4 independent

11. 5x — 3y = —12 consis;:er.,.., 17. 3x — 5y = —15 :15, 5); consistent, 2x + 3y = — 9 ':?de p endent 3x + 5y = 45 independent

12. Zr — 7y = 21 18. x + y = —4 ;3, —7); consistent, 3x + 7y = 14 :nuep2ndent y = — 7 inci.ependent

13. 3x + 4y = 20 2); consistent, 19. x — 3y = — 15 4); consistent, x + 2y = 8 : i?dependent y = 411,--iependervi

14. x — y = — 4 .3, 7); consistent, 20. 12x + 4y = 8 no solution;

x + 3y = 24 independent 3y = 15 — 9x inconsistent

0- C. Exercises Draw (onclusions about the slopes and y-intercepts of the following types

of systems of linear equations.

21. inconsistent systems The 1ineL: have the same slope but different y-intercepts.

22. dependent systems The lines have the same slope and the same y-intercept.

23. independent systems The iines "nava different ;lopes.

<a t^aiatir^e Review

Solve.

24. x — 5 = 8 x = 13 !4.1]

25. 5x— 11 = 2x + 4 x = 3 [4.3]

26. 3. + 40 = 176 8 [4. 7]

27. 4(x — 3) 8x i( —3 [5.4]

28. 2x + 1 > 19 x > 9 or x < — 10 [5.7]

x + 2y - 10 = 0 3x 2y = 14

2y = -x + 10 2y = -3x + 14

=

1 + 5 y = 2

3 x + 7

consistent; independent

y

x — 2y = 4 3x + 2y = —12 —2y= —x+ 4 2y= 2y=-3x-12

y= -Tx — 2 y= 2 x 6

consistent; independent

y

7.2 USING THE GRAPHING METHOD 279

5x - 3y = -12 2x + 3y = -9 -3y = -5x - 12 3y = 2x 9

5 y - -Tx + 4 y= -32 x - 3

consistent; independent

y

11x + y = 19 y= -11x + 19 0 3x + 4y = 20

4y = -3x + 20

- + 5

y

x + = 8 2y= -x + 8

1 y - -2 x + 4

consistent; independent

t • •

Answers

Chapter 7—Systems of Equations and Inequalities

0 9x - y = -5 -y = -9x - 5

Y - 9x 5

y

- 5y = -30 2x - y= 3 -5y = -x - 30 y= 2x + 3

1 y= -5-x y =2x - 3

y

0 9x - 2y =7 3x - 4y = 14 -2y- -9x + 7 4y - -3x + 14

9 7 -3 7

y=- x -- T

4 7 0 3 ± 12 49 5 35.9

5 3 3 28 9 28 _ 37

15 5 ' 15 - 15 ' 15 -

7 14

26 39 _= 21 28 78 78

7 78

8 = 23 = 2 i 343 73 T

6_2 2_1 = 1 93 3 g-3 62.3 2

333

1 9 • 9 -9 27

663 2 8 221

0 527 1 = \ 144 + 72 -

/527 2 _ 144 + 144 -

/529 _ /23 2 _ 23 V 144 - 12 212

0 2x 7y = 21 3x + 7y= 14 -7y= -2x 21 7y= -3x + 14

2 y = -Tx - 3 y= -73 x + 2

y

x - y = -4 x + 3y = 24 -y= -x - 4 3y = -x + 24

y= x + 4 y= -31 x + 8

0 3x + 4y = 4 x - 2y= 8

4y = -3x + 4 -2y= -x + 8 -3

4 X + 1 y = Tx - 4

consistent; independent

y ••

12x - y= -8 -y=-12x- 8

y= 12x + 8

-7 78

ANSWERS 607

608 ANSWERS

3x 5y= 8 3x = 8 - 5y

8 5 X = - -3-y

2x - 4 = 7

2(-3- - ÷y) - 4y = 7 __16 -

y - 4y 7

16 - 10y- 12y~ 21 -22y = 5

-5 y= 22

33: _± 5 ( -2!): 8

66x - 25 = 176 66x = 201

201 _ 67 66 - 22

35 + 30y= 25x 30y = 25x - 35

2535 , y= 30^

_30 30

5 7 Y = -6- x

10v- 12y = 14 2 7 ,

10x - 12( 5

-Tx - = 14 7 1

10x - 10x + 14 14 14 = 14 True Ans. entire line

32y ± 9 = 20x 32y = 20x- 9

_ 20 9 Y - 32 X 32

_ 5 „ Y - ^ - 9 32

15x - 24y = 7 15x - 241 u ^ 392 ) = 7 15x - 15x + = 7 27 = 7 False

Ans. no solution

(1•10) 7. x + 2y -4

2y -x - 4 y

-21 x 2

Ans. ( )

x 2y < 6 2y < -x + 6

-1 y < 2 X ± 3

40 + 9 \iy2 = V49 y= An (40, 7), (40, -7)

0 12x + 4y = 8 3y = 15 - 9x 4y = -12x + 8 y = 5 - 3x y = -3x + 2 y = -3x + 5

A-P

0 5x - 11 2x + 4 3x = 15 x = 5 8 312 x - 176 160( 132 x + 40)= 5x + 12 = 70 5x = 58

«= 5

0(x 4 - 3) s 8x 4x - 12 _s_ 8x -4x _s 12 x -3 9 2x + >19

(-6-)160

2x + 1 > 19 or 2x + 1 < -19 2x> 18 2x< -20 x > 9 x < -10

0 3x + y = -2 y = -3x - 2

3(4) +y= -2

2x - 8y = 7 2x - 8( -3x - 2) = 7 2x + 24x + 16 = 7 26x = -9

-9 ^- 26

-27 26 Y 2

-27 + 26y = -52 26y = -25

-25 i -9 -25 )

Y = 26 \ 26 , 26 /

x - 5y = 6 x = 5y + 6

x - 5( 2135 - 6 „ 75 6 " 23 = 23x + 75 = 138 23x~ 63

63 x = 23

y2 = x + 9

4x + 3y = 9 4(5y + 6) + 3y= 9 20y + 24 + 3y = 9 23y= -15

-15 23

Ans. ( -2./35 )

4x + y2 = 209 4x + (x + 9) = 209 5x = 200 x = 40

0 3x + y = 5 y -3x + 5

y = 3x - 4

y

x

3x -- 5y = -15 3x -+ 5y = 45 -5y = -3x - 15 5y = -3x + 45

3 y 5 x + 3 y - -53 x + 9

consistent; independent

0 x + y = - 4 y = -x - 4

0 x - 3y = -15 -3y = -x - 15 y = -Tx 5

consistent; independent

-7

y = 4

(-3, 4) v~

< 1 1 A‘Pi —2. 0 2 L+

4 • 10 -I. 0 2.

x

C) [6.10] x < 2 [5.2]

4 I 1 I I m I I>

-2 0 2 4

0 [6.6]

[6.1]

y

(-1 '2)

• 5,1)

Write two equations and solve the system by substitution. x y = 63

x — y = 13

17. The sum of two numbers is 63. and their difference is 13. Find the two y = 63 — x x — (63 — x) = 13

numbers. :=E

x — 63 + x = 13

18. The sum of two numbers is 996. The difference of the larger and twice

2x = 76 the smaller is 33. Find the two numbers. - x = 38

C Exercises Solve each system by the substitution method. 19. (x y) 2 = 36 21. x2 + = 25

x + y = 6 e iEs 2X2 3y2 = 5 -20. x2 y = 5

8x2 -- 2y = 0 (1, 4), (-1

' ye Review

Graph. Use number lines or Cartesian planes as appropriate.

22. 1(5, 1), (-1, 2)) 23. x 5 = 7 r4.11

24. 5 -- x > 3

25. x I = 2 I./.6]

26. y = —5x + 3 27. y> 2 — x

38 + y = 63 y = 25 Ans. (38, 25)

0 x + y = 996 x — 2y = 33 y = 996 — x x — 2(996 — x) = 33

x — 1992 + 2x = 33 3x = 2025 x = 675

Ans. (675, 321)

(X ± y)2 = 36 (6)2 = 36 36 = 36 True Ans. entire line

2+ y = 5 8x2 — 2y = 0 y = 5 — x 2 8x2 — 2(5 — x2) = 0

8x2 — 10 + 2x2 = 0 10x2 = 10 X2 = 1 x = -± 1

1 2 + y = 5 (-1)2 y = 5 1 + y = 5 1 + y = 5 y = 4 y = 4 Ans. (1, 4). ( —1, 4)

x2 + y2 = 25 2x2 — 3y2 = 5 y2 = 25 — x 2 2x2 — 3(25 — x2) = 5

2x2 — 75 + 3x 2 = 5 5x2 = 80 x2 = 16 x =

42 + y2 = 25 (-4)2 + y2 = 25

16 y2 = 25 16 + y2 = 25 y2 9 y2 9

Y = y = -±3 Ans. (4, 3), (4, —3). ( - 4, 3), ( -4, — 3)

7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD 287

675 + y = 996 y = 321

x + y = 6

7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD 287