Download y = 4x y = -x + 5 -y = -x - 2 3.x - y = 7 yeS 2x + 3y = 4 0 0 1.x + y = 3 8

05x -L- 6y = 0
• A. Exercises
Is (-2, 5) a solution to each system?
3. x - y = 7
2x + 3y = 4
1. x + y = 3
2x + y = 1
yeS
2. x 2y = 8
3x + y -= 1
n
0
00
4. 3x + 2y = 4
€. S
5x + 3y = 5
For each system determine whether the given point is the solution.
8. 3x - y =- -7, (-1, 4)
5. x + 3y = 13; (2, -5)
x - 5y --= -19 na
6> + y 7 el
9. 5x + 6y = 0; (6, -5)
6. 2> - 5y = 9, (7, 1)
2x + y = 7 tcs
x + y = 8 vfotS
10.
y
= -2, (-2, -8)
3.
-2)
1;
7. x + 2y =
y -= 4x
3> - 2y = 5 rNO
516) + 6(-5) =
30 - 30 = 0
0 = 0
y = - 2
-8 = -2 False
Ox + y =
5
y = -x +
5
2x + y = 7
0 2(6) + (-5) = 7
12- 5=7
7=7
y = 4x
-8 = 4(-2)
--8 = -8
x-y=1
-y = -x + 1
y=x- 1
B. Exercises
Graph each system of linear equations to find its solution.
16. 4x + y = 7
11. x y = 5
x - 3y 18 13 . x - y = 1 '3 2'
17. x + y = 9
12. x - y = -2
11x + y 19
- y 4
18. 9x - y = -5
13. 6> + y = 7
12x - y -8
y=1
19. x - 5y = -30
14. x + 2y = 10
2x - y = 3
9 y + 2y= -6
0
-4)
x - y = -2
-y = -x - 2
y=x+
2
4x - y = 4
-y= -4x + 4
y = 4x - 4
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
14 student answers should approximate the solution (1-,.
3x + 4y
Currnilati.ve Review
3.
Simplify.
3
12 49
21. 5 + 35 • 9
37
15 ,;12.31
24. 6-2 • 2-i
14 1
39
-3 ,'*-1
25.
22.
I
r 266
1 8
• N 343
9- 3-
3
527
1
144 + 72
3
'
31
-
7
275
7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING :WWW.F7•,
x + 2y= 10
2y= -x 10
-1
= x
5
01n4 .-;
9x + 2y = -6
2y = -9x - 6
y= 2 x - 3
= 2 61
8
8x - 3y = -12 4x - y = -8
-3y = -8x - 12 -y = -4x -
8
y = Tx + 4
r I !
cot!
y = 4x + 8
8
0
4x + y = 7
y = -4x + 7
3y = 18
-3y = -x + 18
y =x - 6
x -
•
ii
7.1 SOLVING SYSTEMS OF EQUATIONS BY GRAPHING 275
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.
5x + y = 8
— 5x + 8
gx-Fy.--
y =--x+ 4
nnsistent;
independent
y
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 determine 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
Notice that inconsistent means not consistent and independent means not
dependent.
POSSIBLE SOLUTIONS FOR SYSTEMS OF EQUATIONS
Consistent
Inconsistent
Independent
Dependent
finite
infinite
none
graphs
intersect
graphs
coincide
graphs do not
intersect
one
all points
on the line
none
Number of solutions
Graphs
Applied to lines
(number of solutions)
► A. Exercises
graph marked
off by 2's
®+y=8
y —2x + 8
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.
4); consisten',.
1. x + y = 4
3); consistent:
5. x + 2y — 10 = 0
3x + 2y = 14 independent
5x + y = 8 independent
2. x — y = 3 (4. 1); consistent,
6. 8x — 2y = 6 no solution;
4x — y = 15 independent
y 4x + 3 inconsistent
3. 2x + y = 8 :,ntire line; consistent,
7. x — 2y = 4 1—.7,, —3); consistent,
4x + 2y = 16 dependent
3x + 2y = —12 independent
4. 3x + y = 5 i." ;; 5); consistent,
8. x + 2y = 8 ern:Pie line; consistent,
y = —1+ 4 dependent
4x + 3y = 15 independent
4x + 2y = 16
2y= —4x + 16
y = —2x + 8
consistent;
c ependent
y
278
01
► 3x -y= 5
y —3x + 5
4x + 3y = 15
3y = —4x + 15
—4
y= 3 x + 5
CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES
Case 3
1. Lines coincide.
2. Solution is an infinite set.
3. System is consistent and dependent.
Example
x+y=3
2x + 2y = 6
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 dependent 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
$•• B. Exerciies
x + 2y - 10 = 0 3x 2y = 14
2y = -3x + 14
2y = -x + 10
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
x= 4 .'7"
10. x — y = 6
15. 3x + 4y = 4
x — 2y = 8
16. 3x + y =
_
1
5(
y=
=
+ 5
consistent;
independent
..ant
3
2 x +
7
y
;iista t,
— 5y = 30 .dependent
y = 3x — 4 independent
consis;:er.,.., 17. 3x — 5y = —15 :15, 5); consistent,
11. 5x — 3y = —12
2x + 3y = — 9 ':?de p endent
12. Zr — 7y = 21
3x + 7y = 14 :nuep2ndent
2); consistent,
13. 3x + 4y = 20
x + 2y = 8 : i?dependent
14. x — y = — 4 .3, 7); consistent,
x + 3y = 24 independent
3x + 5y = 45 independent
18. x + y = —4 ;3, —7); consistent,
y = — 7 inci.ependent
19. x — 3y = — 15
4); consistent,
y = 411,--iependervi
20. 12x + 4y = 8 no solution;
3y = 15 — 9x inconsistent
8x - 2y = 6
-2y = -8x +
y = 4x - 3
0- C. Exercises
Draw (onclusions about the slopes and y-intercepts of the following types
of systems of linear equations.
y=
4x + 3
6
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.
X
<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
27. 4(x — 3)
28.
8x i(
8 [4. 7]
x — 2y = 4
3x + 2y = —12
2y=
—2y= —x+ 4 2y=-3x-12
—3 [5.4]
y=
2x + 1 > 19 x > 9 or x < — 10 [5.7]
y=
-T x — 2
2
x
6
consistent;
independent
y
7.2 USING THE GRAPHING METHOD 3
2y = -x --
y= 21
x 2y =
8
1
y= 2 x + 4
x+4
(tx
= —3
x =4
incons'stent
y
279
3
x_ y = 6
-y = -x +
y=x-6
6
5x — 5y = 30
—5y = —5x + 30
y=x-6
7.2 USING THE GRAPHING METHOD
279
0
Answers
2x 7y = 21
3x + 7y= 14
-7y= -2x 21 7y= -3x + 14
2
y = -Tx - 3
y= 73 x + 2
y
Chapter 7—Systems of
Equations and Inequalities
9x - 2y =7
3x - 4y = 14
0 -2y- -9x + 7 4y - -3x + 14
9
7
-3
7
y=- x -- T
11x + y = 19
y= -11x + 19
0
3x + 4y = 20
4y = -3x + 20
-
+5
consistent;
independent
t •
•
y
0
9x - y = -5
-y = -9x - 5
Y - 9x 5
12x - y= -8
-y=-12x- 8
12x + 8
y=
0
4
7
3 ± 12 49
5
35.9
5
3
3
28
9 28 _ 37
5 ' 15 - 15 ' 15 - 15
7
14
28
21
26
39 _= 78
78
y
x+ = 8
2y= -x + 8
-1
y - 2 x+4
x - y = -4
-y= -x - 4
-7
78
7
y= x + 4
x + 3y = 24
3y = -x + 24
-1
y= 3 x + 8
3x + 4y = 4
4y = -3x + 4
x - 2y= 8
-2y= -x + 8
78
8 = 23 = 2
i 343
T
73
6_2 2_1 = 1 3
g-3
62.3
333
1 9 • 9 -9
663
2
221
0
93
2
27
8
527 1 =
\ 144 72
+/527
2 _
144 + 144 -
- 5y = -30
-5y = -x - 30
1
y= -5-x
4
y
2x - y= 3
y= 2x + 3
/529 _ /23 2 _ 23
V 144 -
12 212
y =2x - 3
0
-3
X + 1
5x - 3y = -12
2x + 3y = -9
-3y = -5x - 12 3y = 2x 9
5
-2
y - -Tx + 4
y= 3 x - 3
y = Tx - 4
consistent;
independent
y
••
consistent;
independent
y
ANSWERS
607
0
3x + y = 5
y -3x +
0
y = 3x - 4
5
y
12x + 4y = 8
4y = -12x + 8
3x 5y= 8
3x = 8 - 5y
3y = 15 - 9x
y = 5 - 3x
y = -3x + 5
y = -3x + 2
8
5
X = - -3-y
2x -
4=
7
2(-3- - ÷y) - 4y =
16 - - 4y
__
y
AP
x
7
7
16 - 10y- 12y~ 21
-22y = 5
-5
y= 22
2!):
3 : _± 5 ( -
3x -- 5y = -15 3x -+ 5y = 45
-5y = -3x - 15 5y = -3x + 45
3
y - -53 x + 9
y 5 x + 3
consistent;
independent
0
8
5x - 11
3x = 15
x=5
2x +
4
66x - 25 = 176
66x = 201
_ 67
66 - 22
201
176
160( 321 x + 40)= (-6-)160
312 x
-
2535
, _
30
30
y= 30^
7
5
Y = -6- x
«= 5
0(x
4 - 3) s 8x
4x - 12 _s_ 8x
-4x _s 12
x -3
2x + >19
2x + 1 > 19 or
2x> 18
x > 9
9
0
-7
0 3x +
10v- 12y = 14
7,
2 5
=
10x - 12(-Tx -
7
2x + 1 < -19
2x< -20
x < -10
2x - 8y = 7
2x - 8( -3x - 2) = 7
2x + 24x + 16 = 7
26x = -9
y = -2
y = -3x - 2
-9
^- 26
-2
2
26
Y
-27 + 26y = -52
26y = -25
0
i
-25
Y = 26
x x=
-9 -25 )
\ 26 , 26 /
3y = 9
4(5y + 6) + 3y= 9
20y + 24 + 3y = 9
23y= -15
5y = 6
5y + 6
4x +
consistent;
independent
-15
x-
5(
2 135
-
23
6
„ 75 " 23 = 6
(-3, 4)
75 = 138
23x~ 63
23x +
v~
x = 63
23
y2 = x +
Ans. (
9
-2./35
)
4x + y2
= 209
4x + (x + 9) = 209
5x = 200
x = 40
40 + 9
V49
\iy2 =
y=
608
ANSWERS
10x - 10x + 14 14
14 = 14 True
Ans. entire line
32y ± 9 = 20x
32y = 20x- 9
_ 20
9
Y - 32 X 32
_5„
^ - 32
9
Y-
15x - 24y = 7
15x - 241 u ^
15x - 15x +
392 ) =
7
=7
= 7 False
Ans. no solution
-27
y=4
14
1
27
3 (4) +y=
x - 3y = -15
-3y = -x - 15
y = -Tx 5
)
Ans. (
35 + 30y= 25x
30y = 25x - 35
5x + 12 = 70
5x = 58
x + y = -4
y = -x - 4
8
An
(40, 7), (40, -7)
(1•10)
7. x + 2y
-4
2y -x - 4
-1
2 x 2
y
2y < 6
2y < - x +
x
y<
6
2X±3
-1
x y = 63
y = 63 — x
Write two equations and solve the system by substitution. 17. The sum of two numbers is 63. and their difference is 13. Find the two
numbers. :=E
18. The sum of two numbers is 996. The difference of the larger and twice
the smaller is 33. Find the two numbers.
C Exercises
Solve each system by the substitution method.
21. x2 +
19. (x y)2 = 36
e iEs
2X2
x + y = 6
20. x2 y = 5
8x2 -- 2y = 0 (1, 4), (-1
'
ye
0
= 25
3y2 =
5-
Review
<
6
A‘Pi
1 1
— 2.
Ans. (38, 25)
x — 2y = 33
x + y = 996
y = 996 — x x — 2(996 — x) = 33
x — 1992 + 2x = 33
3x = 2025
x = 675
675 + y = 996
Ans. (675, 321)
y = 321
x+y=
Graph. Use number lines or Cartesian planes as appropriate.
25.
x I = 2 I./.6]
22. 1(5, 1), (-1, 2))
26. y = —5x + 3
23. x 5 = 7 r4.11
0 2 L+
(X ± y)2 = 36
(6)2 = 36
36 = 36 True
Ans. entire line
27. y> 2 — x
24. 5 -- x > 3
38 + y = 63
y = 25
x — y = 13
x — (63 — x) = 13
x — 63 + x = 13
2x = 76
x = 38
• 10
4
8x2 — 2y = 0
2+ y = 5
y = 5 — x 2 8x2 — 2(5 — x2 ) = 0
8x2 — 10 + 2x2 = 0
10x2 = 10
X2 = 1
x = -± 1
-I. 0 2.
(-1)2 y =
1 2 + y = 5
1+y=5
1 + y = 5
y=4
y = 4
Ans. (1, 4). ( —1, 4)
5
x2 + y2 = 25
2
y2 = 25 — x
2x2 — 3y2 = 5
2x2 — 3(25 — x2 ) = 5
2x2 — 75 + 3x 2 = 5
5x2 = 80
x2 = 16
x=
42 + y 2 = 25
16 y2 = 25
(-4)2 + y2 = 25
16 + y2 = 25
y2
y2
9
9
y = -±3
Y =
Ans. (4, 3), (4, —3). ( - 4, 3), ( - 4, — 3)
287
7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD [6.1]
x < 2 [5.2]
4
y
I
-2
1
I
0
I
m
2
I
I>
)
C
[6.10]
4
0 [6.6]
(-1 '2)
5,1)
•
x
7.4 SOLVING SYSTEMS BY THE SUBSTITUTION METHOD
287