10.4 OBJECTIVES - Innovative Learning · PDF fileIn exercises 33 to 40, solve the systems of inequalities graphically. (Note: These have already been graphed as systems of equations

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Graphing Nonlinear Systems10.4

10.4 OBJECTIVES

1. Graph a system of nonlinear equations2. Find ordered pairs associated with the solution set

of a nonlinear system3. Graph a system of nonlinear inequalities4. Use substitution to find the solution set for a non-

linear 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 correspondingto those equations, and then recording the point of intersection. That point represented thesolution to the system of equations. We will use a similar method to find the solution set fora 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.

Zero Solutions One Solution

y

x

y

x

Two Solutions Three Solutions Four Solutions

y

x

y

x

y

x

For the remainder of this section, we will restrict our discussion to a system that has as itsgraph 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

y � 6

796 CHAPTER 10 GRAPHS OF CONIC SECTIONS

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C H E C K Y O U R S E L F 1


y � x2 � 5x � 4

y � 10

First, we will graph the system. From this graph we will be able to see the number ofsolutions. The graph will also give us a way to check the reasonableness of our algebraicresults.

Let’s use the method of substitution to solve the system. Substituting 6, from the secondequation, 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 eitherequation to solve for y, but we know from the second equation that y � 6. The solution setis �(�1, 6), (4, 6)�. Looking at the graph, we see that this is a reasonable solution set for thesystem.

x

y

Solving a Nonlinear System


y � x2 � x � 3

y � 7

Example 2

NOTE Use your calculator toapproximate the solutions forthe system.

Of course, not every quadratic expression is factorable. In Example 2, we must use thequadratic formula.

GRAPHING NONLINEAR SYSTEMS SECTION 10.4 797©

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Let’s look at the graph of the system.

We see two points of intersection, but neither seems to be an integer value for x. Let’s solvethe 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.

The two points of intersection are It is difficult to

check these points against the graph, so we will approximate them. The approximatesolutions (to the nearest tenth) are (�2.6, 7) and (1.6, 7). The graph indicates that these arereasonable answers.

��1 � 117

2, 7� and ��1 � 117

2 �.

x ��1 � 11 � 16

2�

�1 � 117

2

y

x



y � x2 � x � 5

y � 8

As was stated earlier, not every system has two solutions. In Example 3, we will see asystem with no real solution.

Solving a System of Nonlinear Equations


y � x2 � 2x � 1

y � �2

Example 3


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y � x2 � 3x � 5

y � 2

As we did with the previous systems, we will first look at the graph of the system.

Using the method of substitution, we get

�2 � x2 � 2x � 1

0 � x2 � 2x � 3

Using the quadratic formula, we can confirm that there are no real solutions to this system.

�(�2) � 2(�2)2 � 4(1)(3)

2(1)�

2 � 1�8

2

y

x

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.

x

y


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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 � 253x2 � y 2 � 114x2 � 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

y � �4

The other two pairs in the solution set are (�3, �4) and (�3, 4).

We could approximate the solutions, then check those approximations by substitution. Buthow could we find the solutions algebraically? Example 4 illustrates the eliminationmethod.


The solutions set is �(�3, �4),(�3, 4), (3, �4), (3, 4)�

x

y

(�3, 4)

(�3, �4)

(3, 4)

(3, �4)

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Solve by the elimination method.

x2 � y2 � 5

2x2 � 3y 2 � 14

Example 5

Solving a System of Nonlinear Inequalities

Solve the following system.

y � x2 � 3x � 2

y � 6

From Example 1, we have the graph of the relatedsystem of equations.

x

y

Recall that a system of inequalities has as its solutions the set of all ordered pairs that makeevery inequality in the system a true statement. We almost always express the solutions toa system of inequalities graphically. We will do the same thing with nonlinear systems.


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y � x2 � 5x � 4

y � 10

NOTE The solution set is theshaded area above theparabola and below the line.

The first inequality has as its solution set every ordered pair with a y value that is greaterthan (above) the graph of the parabola. The second statement has as its solution set everyordered pair with a y value that is less than (below) the graph of the line. The solution setto the system is the set of ordered pairs that meet both of those criteria. Here is the graphof the solution set.

y

x

Example 6 demonstrates that, even if the related system of equations has no solution, thesystem of inequalities could have a solution.

Example 6

Solving a System of Nonlinear Inequalities


y � x2 � 2x � 1

y � �2

As we did with the previous systems, we will first look at the graph of the related systemof equations (from Example 3.)

y

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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.

x

y

y � �2



y � x2 � 3x � 5

y � 2

C H E C K Y O U R S E L F A N S W E R S

1. �(�1, 10), (6, 10)� 2. � �(1.3, 8), (�2.3, 8)�

3. No real solution 4. �(�1, �3), (�1, 3), (1, �3), (1, 3)�5. 6.

x

y

y � 2

y

x

y � 10

��1 � 113

2, 8�

NOTE The solution continuesbeyond the borders of the grid.

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Exercises

In exercises 1 to 8, the graph of a system of equations is given. Determine how many realsolutions each system has.1. 2.

3. 4.

5. 6.

7. 8. y

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y

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yy

x

y

x

y

x

y

x

y

x

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

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In exercises 9 to 12, draw the graph of a system that has the indicated number of solutions.Use the conic sections indicated.

9. 0 solutions: (a) use a circle and an ellipse, and (b) use a parabola and a line.

(a) (b)

10. 1 solution: (a) use a parabola and a circle, and (b) use a line and an ellipse.

(a) (b)

11. 2 solutions: (a) use a parabola and a circle, and (b) use an ellipse and a parabola.

(a) (b) y

x

y

x

y

x

y

x

y

x

y

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ANSWERS

9.

10.

11.

804

12. 4 solutions: (a) use a circle and an ellipse, and (b) use a parabola and a circle.

(a) (b)

In exercises 13 to 24, graph each system and estimate the solutions.

13. y � x2 � x � 2 14. y � x2 � 3x � 2

y � 4 y � 6

15. y � x2 � 5x � 7 16. y � x2 � 8x � 18

y � 3 y � 6

y

x

y

x

x

y

x

y

y

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sANSWERS

12.

13.

14.

15.

16.

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17. y � x2 � 4x � 7 18. y � x2 � 6x � 7

y � 4 y � 2

19. y � x2 � x � 5 20. y � x2 � 8x � 17

y � 6 y � 5

21. y � x2 � 7x � 11 22. y � x2 � 2x � 2

y � 6 y � 6

x

y

x

y

x

y

x

y

x

yy

x

ANSWERS

17.

18.

19.

20.

21.

22.

806

23. y � x2 � 5 24. y � x2 � 4x � 9

y � 4 y � 2

In exercises 25 to 32, solve using algebraic methods. (Note: These exercises have beensolved graphically in exercises 13 to 24.)

25. y � x2 � x � 2 26. y � x2 � 3x � 2 27. y � x2 � 5x � 7

y � 4 y � 6 y � 3

(See exercise 13.) (See exercise 14.) (See exercise 15.)

28. y � x2 � 8x � 18 29. y � x2 � x � 5 30. y � x2 � 8x � 17

y � 6 y � 6 y � 5

(See exercise 16.) (See exercise 19.) (See exercise 20.)

31. y � x2 � 5 32. y � x2 � 4x � 9

y � 4 y � 2

(See exercise 23.) (See exercise 24.)

In exercises 33 to 40, solve the systems of inequalities graphically. (Note: These havealready been graphed as systems of equations in exercises 13 to 24.)

33. y � x2 � x � 2 34. y � x2 � 3x � 2

y � 4 y � 6


x

y

x

y

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y

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23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

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35. y � x2 � 5x � 7 36. y � x2 � 8x � 18

y � 3 y � 6


37. y � x2 � 4x � 7 38. y � x2 � 6x � 7

y � 4 y � 2


39. y � x2 � 5 40. y � x2 � 4x � 9

y � 4 y � 2


x

y

x

y

x

y

x

y

x

y

x

y

ANSWERS

35.

36.

37.

38.

39.

40.

808

In exercises 41 to 44, (a) graph each system and estimate the solution, and (b) usealgebraic methods to solve each system.

41. y � x2 42. y � x2 � 6x

x � y � 2 3x � y � 4

43. x2 � y2 � 5 44. x2 � y2 � 9

�3x � 4y � 2 x � y � �3

Solve the following applications.

45. The manager of a large apartment complex has found that the profit, in dollars, isgiven by the equation

P � 120x � x2

in which x is the number of apartments rented. How many apartments must be rentedto produce a profit of $3600?

46. The manager of a bicycle shop has found that the revenue (in dollars) from the saleof x bicycles is given by the following equation.

R � x2 � 200x

How many bicycles must be sold to produce a revenue of $12,500?

x

y

x

y

x

y

x

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41.

42.

43.

44.

45.

46.

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47. Find the equation of the line passing through the points of intersection of the graphsy � x2 and x2 � y 2 � 90.

48. Write a system of inequalities to describe the following set of points: The points arein the interior of a circle whose center is the origin with a radius of 4, and above theline y �2.

49. We are asked to solve the following system of equations.

x2 � y � 5

x � y � �3.

Explain how we can determine, before doing any work, that this system cannot havemore than two solutions.

50. Without graphing, how can you tell that the following system of inequalities has nosolution?

x2 � y2 9

y � 4

Solve the following systems algebraically.

51. x2 � y2 � 17 52. x2 � y2 � 29

x2 � y2 � 15 2x2 � y2 � 46

53. x2 � y2 � 8 54. 2x2 � y2 � 3

2x2 � 3y2 � 20 3x2 � 4y2 � 7

Answers1. 2 3. 1 5. 4 7. 1

9. (a) (b) y

x

y

x

ANSWERS

47.

48.

49.

50.

51.

52.

53.

54.

810

11. (a) (b)

13. (3, 4) and (�2, 4) 15. (4, 3) and (1, 3)

17. (�3, 4) and (�1, 4) 19. (0.6, 6) and (�1.6, 6)

21. (6.2, 6) and (0.8, 6) 23. No solution

y

x

y

x

y

x

y

x

y

x

y

x

y

x

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25. (3, 4) and (�2, 4) 27. (4, 3) and (1, 3)

29. or (0.618, 6) and (�1.62, 6)

31. No solution

33. 35.

37. 39.

41. (1, 1) and (�2, 4) 43. (�2, �1) and

45. 60 47. y � 9 49. 51. �(�4, �1), (�4, 1), (4, �1), (4, 1)�

53. �(�2, �2), (�2, 2), (2, �2), (2, 2)

y

x

y

x

�38

25,

41

25�

y

x

y

x

y

x

y

x

��1 � 15

2, 6� and ��1 � 15

2, 6�

812

Documents

10.4 OBJECTIVES - Innovative Learning · PDF fileIn exercises 33 to 40, solve the systems of inequalities graphically. (Note: These have already been graphed as systems of equations