Section 12.4 Solving Nonlinear Systems of Equations
255
12.4 Solving Nonlinear
Systems of Equations
Exercises
2. 2(1)
(2)
7
35
yx
yx
=+
=+
Use substitution.
4. 22
2
(1)
(2)
16
4
xy
xy
+=
=−
Use substitution.
(
)
)
2
22
416
yy
−+=
6. 2
2
8
4
xy
xy
=−
=− +
Use substitution.
8. 22 (1)
(2)
25
5
xy
xy
+=
+=
Solve equation (2) for x, then substitute the
expression for x in equation (1) and solve for y.
55xy x y+=⇒=−
(
)
22
22
2
525
25 10 25
2100
yy
yy y
yy
−+=
−++=
−=
(2)
2
xy
=
Solve equation (2) for y, then substitute the
expression for x in equation (1) and solve for x.
12
2
xyyx=⇒=
2
3
y
=−
Solutions:
()
24
,,2,4
33
⎛⎞
⎟
⎜−−
⎟
⎜⎟
⎜
⎝⎠
Chapter 12 Conic Sections
256
12. (continued)
22
22
30 160
316
yy
yy
−= + =
==−
14. 22
22
2
2
(1)
(2)
430
515
9 45
5
)
)
xy
xy
x
x
+=
−=
=
=
16.
()
1
22 22
22 22
2
(1)
(2)
20 20
432 432
3 12
xy xy
xy xy
y
×−
+=⎯⎯⎯→− − =−
+=⎯⎯→+=
=
18.
()
22 22
1
22 22
2
28 40 28 40
5 8 16 5 8 16
xy xy
xy xy
×−
+=⎯⎯→+=
+=⎯⎯⎯→− − =−
2
2
856
7
7
y
y
y
=
=
=±
Solutions:
(
)
(
)
22, 7, 22,7,
ii
−− −
2
2
2
4832
440
10
10
x
x
x
x
−=
=
=
=±
Section 12.4 Solving Nonlinear Systems of Equations
257
22. (continued)
22
272
xy
+=
Solutions:
(
)
(
)
6, 0 , 6, 0−
24.
22
27 22
19327
39
2
xy xy
x
×
+=⎯⎯→ + =
=±
22
2
2
2
2
64 24
62 4 24
12 4 24
412
3
3
xy
y
y
y
y
y
+=
⋅+ =
+=
=
=
=±
26.
2
9
222
22
16 22
2
2
199
9
1416
16 4
5 25
5
xyxy
xy xy
y
y
×
×
−+=⎯⎯→− + =
−=⎯⎯→− =
=
=
28.
(
)
2
2
(1)
19
xy
++ =
(
)
24 2
42
22
699
70
70
xx x
xx
xx
++ +=
+=
+=
Solutions:
(
)
(
)
(
)
0,2 , 7, 5 , 7, 5ii−− −
30.
()
1
22 22
22 22
2
2
13 13
321 321
2 8
4
xy xy
xy xy
x
x
×−
+=⎯⎯⎯→− − =−
+=⎯⎯→+=
=
=
22
413 413
99
33
yy
yy
yy
+= +=
==
=± =±
Solutions:
(
)
(
)
(
)
(
)
2, 3 , 2, 3 , 2, 3 , 2,3−− − −
32. 2
(1)
7
yx
=−
Chapter 12 Conic Sections
258
34.
)
)
22
18 22
22
18 22
1618
18 3
12318
96
xy xy
xy xy
×
×
+=⎯⎯→ + =
+=⎯⎯→ + =
36.
22
2
2
40.5 1.5 2
0.5 2.5 2 0
540
tt t t
tt
tt
−+ = + +
−+=
−+=
38.
()
22
22
4
22
410,000
4 10,000
410,000
xy
xy
xy
×−
+= ⎯⎯⎯→
+=
+= ⎯⎯⎯→
38. (continued)
The pathways intersect at the points
(
)
(
)
20 5, 20 5 , 20 5, 20 5 ,
205, 205, and205,205.
−− −
−
62413
6382
19 3
hk
hk
kh
−−+=
−+ =
=−
2
2
()() ()()
()
2
57 57 4 5 160
25
57 49
h
h
−− ± − −
=
±
=
Section 12.5 Solving Nonlinear Inequalities and Nonlinear Systems of Inequalities
259
Mindstretchers
1. 22
9
x
xy
ye
+=
=
(
)
(
)
1.04, 2.82 , 3.00, 0.05−
The asymptotes of the hyperbola are
42.
2
yxx=± =±
Solving the system gives us
22
416
2
xy
yx
−=
=
3. 22
22
22
(1)
(2)
(3)
15
222
313
xy
xy
xy
+=
+=
−=
3. (continued)
Check these values in 22
15.xy+=
(
)
(
)
22
22
77, 228xy=± = =± =
22
15
7815
xy+=
+=
12.5 Solving Nonlinear
Inequalities and Nonlinear
Systems of Inequalities
Exercises
(
)
22
The corresponding equation is:
4
Circle with radius = 2, center: 0,0
xy
+=
Chapter 12 Conic Sections
260
8.
(
)
(
)
()()
22
22
321
The corresponding equation is:
321
xy
xy
++−<
++−=
10.
(
)
2
24yxx<−
()
2
2
22
2
The corresponding equation is:
28
24
44
24 2
22
yx x
yxx
yxx
=−
=−
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=−+ −
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠
⎜⎟
⎜
⎝⎠
12. 22
16 64xy+>
The corresponding equation is:
22
22
16 64
xy
+=
14. 22
25 9 225xy+≤
The corresponding equation is:
22
22
25 9 225
1
xy
xy
+=
+=
the region inside the ellipse.
Section 12.5 Solving Nonlinear Inequalities and Nonlinear Systems of Inequalities
261
16. 22
2832yx−<
The corresponding equation is:
)
)
22
28 32
yx
−=
?
?
200 0 32
200 32 False
−<
<
(
)
Do not shade the region containing 0,10 .
(
)
Region II, “between” the two curves.
Test point is 0,0 .
() ()
?
22
20 80 32
−<
18. 25
The corresponding equation is:
yx≤− +
2
5
yx
=− +
(
)
(
)
22
50, 5, 5,
-intercepts are 5,0 and 5,0 .
xxx
x
−+= = =±
−
(
)
()
(
)
???
2
The boundary is solid. Test point is 0,0 .
0 0 5, 00 5, 05 True.
Shade the region containing 0,0 .
≤− + ≤ + ≤
11
yx
≤−
()
?
?
1
001
2
01 False
≤−
≤−
(
)
Shade the region not containing 0,0 .
Chapter 12 Conic Sections
262
18. (continued)
20. 22
22
16
The corresponding equation is:
16
xy
xy
+>
+=
(
)
Circle, center 0,0 , radius 4.
()
)
)
22
22
4
The corresponding equation is:
4
Circle center 0,0 , radius 2.
xy
xy
+<
+=
22. 221yx x<++
The corresponding equation is:
(
)
2
221 1 0yx x y x=++⇒=++
(
)
Parabola opening up with vertex at 1,0−
() () ( )
() () ( )
() () ( )
2
2
2
0020110,1
1121141,4
2222192,9
++=
++=
++=
Boundary is dashed. Test point is
(
)
0, 0 .
()
2
2
24
22
13
yxx
yx
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− + + − +
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠⎜⎟
⎜
⎝⎠
=− + −
(
)
Parabola opening down, vertex at 1, 3 .−−
()
2
24 ,
xyxx xy=− − −
(
)
Shade the area not containing 0, 0 .
The solution is the area enclosed in the
downward opening parabola.
Section 12.5 Solving Nonlinear Inequalities and Nonlinear Systems of Inequalities
263
24. 22
12 27 108xy+>
The corresponding equation is:
22
22
12 27 108
1
94
xy
xy
+=
+=
)
3, 2
ab==
()()() ()
(
)
Ellipse passing through
1, 0 , 1, 0 , 0, 5 , and 0, 5
Boundary is dashed. Test point is 0,0 .
−−
() ()
?
22
?
25 0 0 25
025 True
+<
<
(
)
Shade the area containing 0,0 ,
the inside of the ellipse.
26. 22
16 4 64xy+>
The corresponding equation is:
22
22
16 4 64
1
416
xy
xy
+=
+=
2, 4
ab==
() ()
(
)
Hyperbola passing through
0, 6 , and 0, 6 with asymptotes
6 or 3 and 3 .
2
Boundary is solid. Test point is 0,0 .
yxyxyx
−
=± = =−
() ()
?
22
?
09036
036 True
−≤
≤
Chapter 12 Conic Sections
264
28. 223xy y>−+
The corresponding equation is:
()
(
)
2
2
2
23
2131
12
xy y
xy y
xy
=−+
=−++−
=− +
(
)
Parabola opening right, vertex at 2,1
()
2
23 ,
yxyy xy=−+
Boundary is dashed. Test point is
(
)
0, 0 .
()
?2
?
00 20 3
03 False
>− +
>
(
)
Shade the area not containing 0, 0 .
223xy y≥− + −
(
)
Parabola opening left, vertex at 2,1−
)
)
()
() () ( )
2
2
23 ,
1121366,1
yxyy xy=− + −
−−−+−−=− −−
28. (continued)
The combined areas, not containing
(
)
0,0 and containing
(
)
0,0 do overlap.
The solution is the area enclosed in the
right opening parabola.
30. 223yx x≤− + +
The corresponding equation is:
()
2
22
2
2
23
22
23
22
14
yx x
yxx
yx
=− + +
⎛⎞
⎛⎞ ⎛⎞
−−
⎟
⎜⎟⎟
⎟
⎜⎜
⎜
=− − + + +
⎟⎟
⎟
⎜⎜
⎜⎟⎟
⎟
⎜⎜
⎝⎠ ⎝⎠⎜⎟
⎜
⎝⎠
=− − +
(
)
Parabola opening down, vertex at 1, 4 .
() () ( )
() () ( )
2
2
2222332,3
3323303,0
−+ +=
−+ +=
Boundary is solid. Test point is
(
)
0, 0 .
Section 12.5 Solving Nonlinear Inequalities and Nonlinear Systems of Inequalities
265
32. 22
936xy−≤
The corresponding equation is:
22
936
xy
−=
2
Boundary is solid.
There are three regions.
(
)
Region I, the left region.
Test point is 3,0 .−
()()
?
22
)
)
)
)
93 0 36
−− ≤
() ()
?
22
?
93 0 36
81 36 False
−≤
≤
(
)
Do not shade the region containing 3, 0 .
34. 22
4xy+≤
The corresponding equation is:
22
4
xy+=
the inside of the circle.
() ( )
() ( )
() ( )
2
2
2
00.3099 0,9
10 0.3 10 9 39 10,39
20 0.3 20 9 129 20,129
+=
+=
+=
Boundary is solid. Test point is
(
)
0, 0 .
Chapter 12 Conic Sections
266
36. a. (continued)
The corresponding equation is:
)
2
2
12 0.2
Rx x
=−
(
)
Parabola opening down, vertex at 30,180
() ( )
() () ( )
2
2
12 0.2 ,
0 120 0.20 0 0,0
xyRx x x xy==−
−=
Boundary is solid. Test point is
(
)
10,0 .
)
() ()
?2
01210 0.210
≤−
38.
22
1
400 225
xy
+≥
The corresponding equation is:
22
22
?
00
1
400 225
01 False
+≥
≥
(
)
Shade the region not containing 0,0 ,
()()( )()
(
)
Ellipse passing through
25,0 , 25, 0 , 0, 20 , and 0, 20
Boundary is solid. Test point is 0,0 .
−−
ellipses.
Section 12.5 Solving Nonlinear Inequalities and Nonlinear Systems of Inequalities
267
Mindstretchers
1. Yes, the system
)
(
)
2
2
39
xy
++ ≤
2. 22
936xy+≥
The corresponding equation is:
22
936
xy
+=
The corresponding equation is:
22
22
416
1
16 4
xy
xy
−=
−=
)
)
4, 2
Hyperbola passing through
ab
==
2. (continued)
24xy−>
The corresponding equation is:
24
xy
−=
()
?
?
020 4
04 False
−>
>
(
)
Shade the region not containing 0,0 .
3. The region is inside the ellipse whose
boundary is solid. The ellipse passes
through
(
)
(
)
(
)
(
)
3, 0 , 3, 0 , 0, 5 , and 0, 5 ,−−
so 3, 5.ab==
The corresponding equation is:
22
1
925
xy
+=
Chapter 12 Conic Sections
268
3. (continued)
The region is below the parabola whose
vertex is
(
)
0,4 and passes through (2,0).
The boundary is solid.
)
the relation (
?
R
) must hold for
(
)
0, 1 .−
()
?2
?
104
14
14
−≤− +
−≤
−≤
)
)
The corresponding equation is: yx=−
Since the region contains the point
(
)
0, 1 ,−
the relation (
?
R
) must hold for
(
)
0, 1 .−
3. (continued)
The three inequalities are:
22
1
925
xy
+≤