10.4
Graphing Nonlinear Systems
10.4
OBJECTIVES
1. Graph a system of nonlinear equations
2. Find ordered pairs associated with the solution set
of a nonlinear system
3. Graph a system of nonlinear inequalities
4. Use substitution to find the solution set for a nonlinear system.
5. Identify the solution set of a system of nonlinear
inequalities
In Section 5.1, we solved a system of linear equations by graphing the lines corresponding
to those equations, and then recording the point of intersection. That point represented the
solution to the system of equations. We will use a similar method to find the solution set for
a nonlinear system. A system with two or more conic curves can have zero, one, two, three,
or four solutions. The following graphs represent each of those possibilities.
y
y
x
x
Zero Solutions
y
One Solution
y
x
x
Two Solutions
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y
Three Solutions
x
Four Solutions
For the remainder of this section, we will restrict our discussion to a system that has as its
graph a line and a parabola. Such a system has either zero, one, or two solutions.
Example 1
Solving a System of Nonlinear Equations
Solve the following system of equations.
y x2 3x 2
y6
795
796
CHAPTER 10
GRAPHS OF CONIC SECTIONS
First, we will graph the system. From this graph we will be able to see the number of
solutions. The graph will also give us a way to check the reasonableness of our algebraic
results.
y
NOTE Use your calculator to
approximate the solutions for
the system.
x
Let’s use the method of substitution to solve the system. Substituting 6, from the second
equation, for y in the first equation, we get
6 x2 3x 2
0 x2 3x 4
0 (x 4)(x 1)
The x values for the solutions are 1 and 4. We can substitute these values for x in either
equation to solve for y, but we know from the second equation that y 6. The solution set
is (1, 6), (4, 6). Looking at the graph, we see that this is a reasonable solution set for the
system.
CHECK YOURSELF 1
Solve the following system of equations.
y x2 5x 4
y 10
Example 2
Solving a Nonlinear System
Solve the following system of equations.
y x2 x 3
y7
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Of course, not every quadratic expression is factorable. In Example 2, we must use the
quadratic formula.
GRAPHING NONLINEAR SYSTEMS
SECTION 10.4
797
Let’s look at the graph of the system.
y
x
We see two points of intersection, but neither seems to be an integer value for x. Let’s solve
the system algebraically. Using the method of substitution, we find
7 x2 x 3
0 x2 x 4
The result is not factorable, so we use the quadratic formula to find the solutions.
x
1 11 16
1 117
2
2
1 117
1 117
, 7 and
. It is difficult to
2
2
check these points against the graph, so we will approximate them. The approximate
solutions (to the nearest tenth) are (2.6, 7) and (1.6, 7). The graph indicates that these are
reasonable answers.
The two points of intersection are
CHECK YOURSELF 2
Solve the following system of equations.
y x2 x 5
y8
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As was stated earlier, not every system has two solutions. In Example 3, we will see a
system with no real solution.
Example 3
Solving a System of Nonlinear Equations
Solve the following system of equations.
y x2 2x 1
y 2
CHAPTER 10
GRAPHS OF CONIC SECTIONS
As we did with the previous systems, we will first look at the graph of the system.
y
x
Using the method of substitution, we get
2 x2 2x 1
0 x 2 2x 3
Using the quadratic formula, we can confirm that there are no real solutions to this system.
2 18
(2) 2(2)2 4(1)(3)
2(1)
2
CHECK YOURSELF 3
Solve the following system of equations.
y x2 3x 5
y2
Consider the system consisting of the following two equations:
x2 y2 25
3x2 y2 11
The graph of the system indicates there are four solutions.
y
x
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798
GRAPHING NONLINEAR SYSTEMS
SECTION 10.4
799
We could approximate the solutions, then check those approximations by substitution. But
how could we find the solutions algebraically? Example 4 illustrates the elimination
method.
Example 4
Solving a Nonlinear System by Elimination
Solve the following system algebraically.
x2 y 2 25
3x2 y 2 11
As was the case with linear systems, we can eliminate one of the variables. In this case,
adding the equations eliminates the y variable.
x2 y 2 25
3x2 y 2 11
4x2
36
Dividing by 4, we have
x2 9, so
x 3
Substituting the value 3 into the first equation
(3)2 y 2 25
9 y 2 25
y 2 16
y 4
Two of the ordered pairs in the solution set are (3, 4) and (3, 4).
Substituting the value 3 into the first equation
(3)2 y 2 25
9 y 2 25
y 2 16
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y 4
The other two pairs in the solution set are (3, 4) and (3, 4).
CHAPTER 10
GRAPHS OF CONIC SECTIONS
The solutions set is (3, 4),(3, 4), (3, 4), (3, 4)
y
(3, 4)
(3, 4)
x
(3, 4)
(3, 4)
CHECK YOURSELF 4
Solve by the elimination method.
x2 y2 5
2x2 3y 2 14
Recall that a system of inequalities has as its solutions the set of all ordered pairs that make
every inequality in the system a true statement. We almost always express the solutions to
a system of inequalities graphically. We will do the same thing with nonlinear systems.
Example 5
Solving a System of Nonlinear Inequalities
Solve the following system.
y x2 3x 2
y6
From Example 1, we have the graph of the related
system of equations.
y
x
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800
GRAPHING NONLINEAR SYSTEMS
SECTION 10.4
801
The first inequality has as its solution set every ordered pair with a y value that is greater
than (above) the graph of the parabola. The second statement has as its solution set every
ordered pair with a y value that is less than (below) the graph of the line. The solution set
to the system is the set of ordered pairs that meet both of those criteria. Here is the graph
of the solution set.
y
NOTE The solution set is the
shaded area above the
parabola and below the line.
x
CHECK YOURSELF 5
Solve the following system.
y x2 5x 4
y 10
Example 6 demonstrates that, even if the related system of equations has no solution, the
system of inequalities could have a solution.
Example 6
Solving a System of Nonlinear Inequalities
Solve the following system.
y x2 2x 1
y 2
As we did with the previous systems, we will first look at the graph of the related system
of equations (from Example 3.)
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y
x
802
CHAPTER 10
GRAPHS OF CONIC SECTIONS
The solution set is now the set of all ordered pairs below the parabola (y x2 2x 1)
and above the line (y 2). Here is the graph of the solution set.
y
NOTE The solution continues
beyond the borders of the grid.
x
y 2
CHECK YOURSELF 6
Solve the following system.
y x2 3x 5
y2
CHECK YOURSELF ANSWERS
3. No real solution
5.
y
1 113
, 8 (1.3, 8), (2.3, 8)
2
4. (1, 3), (1, 3), (1, 3), (1, 3)
6.
y
2.
y 10
y2
x
x
© 2001 McGraw-Hill Companies
1. (1, 10), (6, 10)
Name
10.4 Exercises
Section
Date
In exercises 1 to 8, the graph of a system of equations is given. Determine how many real
solutions each system has.
1.
2.
y
ANSWERS
y
1.
2.
x
x
3.
4.
5.
3.
4.
y
6.
y
7.
8.
x
5.
x
6.
y
y
x
x
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7.
8.
y
x
y
x
803
ANSWERS
9.
In exercises 9 to 12, draw the graph of a system that has the indicated number of solutions.
Use the conic sections indicated.
10.
9. 0 solutions: (a) use a circle and an ellipse, and (b) use a parabola and a line.
11.
(a)
(b)
y
y
x
x
10. 1 solution: (a) use a parabola and a circle, and (b) use a line and an ellipse.
(a)
(b)
y
y
x
x
11. 2 solutions: (a) use a parabola and a circle, and (b) use an ellipse and a parabola.
(b)
y
x
804
y
x
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(a)
ANSWERS
12.
12. 4 solutions: (a) use a circle and an ellipse, and (b) use a parabola and a circle.
13.
(a)
(b)
y
y
14.
15.
16.
x
x
In exercises 13 to 24, graph each system and estimate the solutions.
13. y x2 x 2
14. y x 2 3x 2
y4
y6
y
y
x
x
15. y x2 5x 7
16. y x 2 8x 18
y3
y6
y
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y
x
x
805
ANSWERS
17.
17. y x2 4x 7
18. y x2 6x 7
y4
18.
y2
y
19.
y
20.
21.
x
x
22.
19. y x2 x 5
20. y x2 8x 17
y6
y5
y
y
x
21. y x2 7x 11
x
22. y x2 2x 2
y6
y6
y
y
x
© 2001 McGraw-Hill Companies
x
806
ANSWERS
23. y x2 5
23.
24. y x2 4x 9
y4
y2
24.
y
y
25.
26.
27.
x
x
28.
29.
In exercises 25 to 32, solve using algebraic methods. (Note: These exercises have been
solved graphically in exercises 13 to 24.)
30.
25. y x2 x 2
31.
26. y x 2 3x 2
y4
(See exercise 13.)
27. y x 2 5x 7
y6
(See exercise 14.)
y3
(See exercise 15.)
32.
33.
28. y x 8x 18
29. y x x 5
2
30. y x 8x 17
2
y6
(See exercise 16.)
2
y6
(See exercise 19.)
31. y x2 5
y5
(See exercise 20.)
34.
32. y x 2 4x 9
y4
(See exercise 23.)
y2
(See exercise 24.)
In exercises 33 to 40, solve the systems of inequalities graphically. (Note: These have
already been graphed as systems of equations in exercises 13 to 24.)
© 2001 McGraw-Hill Companies
33. y x2 x 2
34. y x2 3x 2
y4
(See exercise 13.)
y6
(See exercise 14.)
y
y
x
x
807
ANSWERS
35.
35. y x2 5x 7
36.
37.
36. y x2 8x 18
y3
(See exercise 15.)
y6
(See exercise 16.)
y
y
38.
39.
40.
x
x
37. y x2 4x 7
38. y x2 6x 7
y4
(See exercise 17.)
y2
(See exercise 18.)
y
y
x
x
39. y x 2 5
40. y x 2 4x 9
y4
(See exercise 23.)
y2
(See exercise 24.)
y
y
x
© 2001 McGraw-Hill Companies
x
808
ANSWERS
In exercises 41 to 44, (a) graph each system and estimate the solution, and (b) use
algebraic methods to solve each system.
41. y x
42. y x 6x
2
41.
42.
2
xy2
43.
3x y 4
44.
y
y
45.
46.
x
43.
x2 y2 5
3x 4y 2
x
44. x 2 y 2 9
x y 3
y
y
x
x
Solve the following applications.
45. The manager of a large apartment complex has found that the profit, in dollars, is
given by the equation
P 120x x 2
in which x is the number of apartments rented. How many apartments must be rented
to produce a profit of $3600?
46. The manager of a bicycle shop has found that the revenue (in dollars) from the sale
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of x bicycles is given by the following equation.
R x 2 200x
How many bicycles must be sold to produce a revenue of $12,500?
809
ANSWERS
47.
47. Find the equation of the line passing through the points of intersection of the graphs
y x 2 and x 2 y 2 90.
48.
48. Write a system of inequalities to describe the following set of points: The points are
in the interior of a circle whose center is the origin with a radius of 4, and above the
line y 2.
49.
50.
49. We are asked to solve the following system of equations.
51.
x2 y 5
52.
x y 3.
53.
Explain how we can determine, before doing any work, that this system cannot have
more than two solutions.
54.
50. Without graphing, how can you tell that the following system of inequalities has no
solution?
x 2 y2 9
y4
Solve the following systems algebraically.
51. x 2 y2 17
52.
x 2 y2 15
53.
x 2 y2 8
2x 2 3y2 20
x 2 y2 29
2x 2 y2 46
54. 2x 2 y2 3
3x 2 4y2 7
Answers
9. (a)
3. 1
5. 4
7. 1
(b)
y
x
810
y
x
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1. 2
11. (a)
(b)
y
y
x
13. (3, 4) and (2, 4)
x
15. (4, 3) and (1, 3)
y
y
x
17. (3, 4) and (1, 4)
x
19. (0.6, 6) and (1.6, 6)
y
y
x
21. (6.2, 6) and (0.8, 6)
23. No solution
y
y
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x
x
x
811
25. (3, 4) and (2, 4)
27. (4, 3) and (1, 3)
1 15
1 15
, 6 and
, 6 or (0.618, 6) and (1.62, 6)
2
2
31. No solution
33.
35.
y
y
x
x
37.
39.
y
y
x
x
43. (2, 1) and
41. (1, 1) and (2, 4)
y
47. y 9
49.
53. (2, 2), (2, 2), (2, 2), (2, 2)
812
38 41
y
x
45. 60
25, 25
x
51. (4, 1), (4, 1), (4, 1), (4, 1)
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29.