Chapter 4
SOLVING SYSTEMS OF
LINEAR EQUATIONS AND
INEQUALITIES
4.1 Introduction to Systems
of Linear Equations:
Solving by Graphing
Exercises
2. A solution of a system of two linear equations
6. a.
(
)
(
)
269 3
254 3
52 3 False
−− −=
−+ =
=
A solution
c.
(
)
(
)
962 3
912 3
21 3 False
−−=
+=
=
8. a.
(
)
(
)
21 2 9 30
218 30
20 30 False
−−=
+=
=
8. (continued)
b.
() ()
() ()
216 21 30
32 2 30
30 30 True
816 21 10
128 2 10
130 10 False
−=
−=
=
+=−
+=−
=−
Not a solution
c.
(
)
(
)
22 2 11 30
422 30
26 30 False
−− =
+=
=
10. a. IV
b. II
c. I
()
1,1
Chapter 4 Solving Systems of Linear Equations and Inequalities
66
16.
18.
20.
22.
24.
26.
28.
30.
Section 4.1 Introduction to Systems of Linear Equations: Solving by Graphing
67
32.
34.
36.
38.
40.
42.
44.
46. a.
() ()
226 2
212 2
10 2 False
264
−− −=−
−+ =−
=−
−−−=
Chapter 4 Solving Systems of Linear Equations and Inequalities
68
46. c.
() ()
10 2 6 2
−=−
48. a. 450
350
xy
xy
+=
−=
b.
50. a. 10p= (Flat ISP)
0.25 2.5ph=+
(Variable ISP)
b.
52.
54.
The washer costs $850 and the dryer costs
$650.
2. No, it is not possible for a system of two
linear equations to have exactly two
solutions. Since two points determine a
straight line, any lines with two common
points will coincide. Such a system will
Section 4.2 Solving Systems of Linear Equations by Substitution
69
4.2 Solving Systems of Linear
Equations by Substitution
Exercises
2. (1)
()
7
53 7
xy
yy
−=
+−=
4. (1) 221
5221
721
3
yx
xx
x
x
=− −
=− −
=−
=−
6. (1)
()
15
415
xy
xx
−=
−− =
6. Check:
(1)
()
31215
312 15
15 15
−− =
+=
=
(2)
()
12 4 3
12 12
−=−
−=−
()
3, 12−
()
530
530
20
0
yy
yy
y
y
−+ =
−+ =
−=
=
3
xy
=
(2)
()
23 6
23 3 6
xy
yy
−=
−=
Chapter 4 Solving Systems of Linear Equations and Inequalities
70
12. Check:
(1)
()
632 0
−=
14. (2) 35
35
xy
yx
+=−
=− −
(1)
()
35 1
3535 1
xy
xx
+=−
+−−=−
Check:
(1)
() ()
32 51 1
65 1
11
−+ =−
−+ =−
−=−
16. (1) 24
24
xy
xy
+=
=− +
18. (1) 31
31
xy
xy
+=
=− +
(2)
()
35 2
33 15 2
xy
yy
−− =−
−− +− =−
18. (continued)
⎛⎞
(1) 11
31
44
131
44
11
⎟⎟
⎜⎜
+=
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝⎠
+=
=
11
,
44
⎜⎟
⎝⎠
20. (1)
()
4252 4
4104 4
10 4
xx
xx
+−=
+− =
=
12 12
−=−
Infinitely many solutions, namely all
ordered pairs that satisfy both equations
6606 60
60 60
vv
−+ + =
=
Infinitely many solutions, namely all
ordered pairs that satisfy both equations
Section 4.2 Solving Systems of Linear Equations by Substitution
71
26. (1) 1
1
ab
ab
−=−
=−
(2)
()
65 1
655
5
bb
bb
b
=−
=−
=−
28. (2) 10 0
10
lw
lw
−− =
=+
36
l
=
Check:
(1)
() ()
236 326 6 0
72 78 6 0
66 0
00
−+=
−+=
−+ =
=
32. (1)
()
512
512
412
3
yy
yy
y
y
−− − =
−=
=
=
(2)
()
53
x
=−
3
yx
=
(1)
()
23 3 6
xx
−=
(1)
() ()
26 32 6
12 6 6
66
−=
−=
=
(2)
()
632 0
66 0
−=
−=
Chapter 4 Solving Systems of Linear Equations and Inequalities
72
38. (1) 260
260
cf
cf
+=
=−
260 246 14−=
40. (1)
(
)
0.2 1 0.5 8
0.2 4
jw
jw
+=
+=
42. (1) 30
30
xy
xy
+=
=−
(2)
0.2 0.04 0.1 30
xy
+=
(1) 18.75 30
1
11.25 11 4
x
x
+=
==
1
11 4 L of the 20% solution should be mixed
44. (1) 3bm=
(2)
()
0.08 0.04 112
0.08 3 0.04 112
bm
mm
+=
+=
0.04 1200 0.08 1120
1200 0.04 1120
0.04 80
xx
x
x
+− =
−=
−=−
Mindstretchers
1. Yes the situation can be modeled by the
system below:
Ch
=+
which the two companies will charge the
same amount. In fact, from the given
information, the second landscaping
company charges both a lower flat fee and
a lower per-hour rate, which means that the
total amount charged by this company will
always be less than the total amount
Section 4.3 Solving Systems of Linear Equations by Elimination
73
4.3 Solving Systems of Linear
Equations by Elimination
Exercises
2. 10
xy
−=
4. 58
21
77
1
xy
xy
x
x
−=
+=−
=
=
()
6. 4
28
ab
ab
+=−
−+ =−
()
5, 2−
10. 82 3
49
xy
xy
+=
+=−
82 3
82 18
xy
xy
+=
−− =
82 6
00
xy
+=−
=
Infinitely many solutions, namely all
ordered pairs that satisfy both equations
()
72 4 6
14 4 6
y
y
−+ =−
−+ =−
Chapter 4 Solving Systems of Linear Equations and Inequalities
74
16. (continued)
(
)
581 7
58 7
515
3
x
x
x
x
−+ =−
−+=−
−=−
=
()
3,1
20. 10 3 9
32 5
xy
xy
−=
−=−
20 6 18
96 15
11 33
3
xy
xy
x
x
−+=−
−=−
−=−
=
()
10 3 3 9
30 3 9
321
7
y
y
y
y
−=
−=
−=−
=
()
3, 7
)
24. 66 3.6
48 16
xy
xy
−=−
−+ =−
24 24 14.4
24 48 96
24 110.4
4.6
xy
xy
y
y
−=−
−+ =−
=−
=−
15 60
4
x
x
=
=
()
34 3 0
12 3 0
312
4
y
y
y
y
−=
−=
−=−
=
()
4, 4
28. 37
42 11
yx
xy
=− +
+=
37
4211
xy
xy
+=
+=
Section 4.3 Solving Systems of Linear Equations by Elimination
75
30. 6
3310
xy
xy
−=
=+
32. 96 3
32 6
xy
xy
−=
−=
34. 74 12
43
pq
pq
+=−
+=−
36. 43 5.5
56 3.5
xy
xy
+=−
+=−
86 11
56 3.5
xy
xy
−− =
+=−
38.
(
)
(
)
214
26
rc
rc
+=
−=
40. 218
22 34
hx
hx
+=
+=
The atomic weight of oxygen is
approximately 16 and that of hydrogen
is approximately 1.
0.375 0.875 35.25
0.5 40
80
sm
m
m
+=
−=−
=
10 17
7
x
x
+=
=
She bought seven $12 scarves and ten
$15 scarves.
46.
52
xy
=
42 40
40
xy
x
−+ =
=
()
540 2
y
=