ch04-6

C R A M E R ’ S R U L E F O R S Y S T E M S I NT H R E E V A R I A B L E S

The solution of linear systems involving three variables using determinants is verysimilar to the solution of linear systems in two variables using determinants.However, you first must learn to find the determinant of a 3 � 3 matrix.

Minors

To each element of a 3 � 3 matrix there corresponds a 2 � 2 matrix that isobtained by deleting the row and column of that element. The determinant ofthe 2 � 2 matrix is called the minor of that element.

E X A M P L E 1 Finding minorsFind the minors for the elements 2, 3, and �6 of the 3 � 3 matrix

� �.

SolutionTo find the minor for 2, delete the first row and first column of the matrix:

� �Now find the determinant of � �:

� � � (�2)(7) � (�6)(3) � 4

The minor for 2 is 4. To find the minor for 3, delete the second row and third columnof the matrix:

� �Now find the determinant of � �:

� � � (2)(�6) � (4)(�1) � �8

The minor for 3 is �8. To find the minor for �6, delete the third row and the secondcolumn of the matrix:

� �2 �1 �80 �2 34 �6 7

2 �14 �6

2 �14 �6

2 �1 �80 �2 34 �6 7

�2 3�6 7

�2 3�6 7

2 �1 �80 �2 34 �6 7

2 �1 �80 �2 34 �6 7

4.6 Cramer’s Rule for Systems in Three Variables (4-39) 237

4.6

I n t h i ss e c t i o n

● Minors

● Evaluating a 3 � 3Determinant

● Cramer’s Rule

Now find the determinant of � �:

� � � (2)(3) � (0)(�8) � 6

The minor for �6 is 6. ■

Evaluating a 3 � 3 Determinant

The determinant of a 3 � 3 matrix is defined in terms of the determinants ofminors.

Determinant of a 3 � 3 Matrix

The determinant of a 3 � 3 matrix is defined as follows:

� � � a1 � � � � a2 � � � � a3 � � �

Note that the determinants following a1, a2, and a3 are the minors for a1, a2,and a3, respectively. Writing the determinant of a 3 � 3 matrix in terms of minorsis called expansion by minors. In the definition we expanded by minors about thefirst column. Later we will see how to expand by minors using any row or columnand get the same value for the determinant.

E X A M P L E 2 Determinant of a 3 � 3 matrixFind the determinant of the matrix by expansion by minors about the first column.

� �Solution

� � � 1 � � � � (�2) � � � � 0 � � �� 1 � [36 � (�42)] � 2 � (27 � 35) � 0 � [18 � (�20)]

� 1 � 78 � 2 � (�8) � 0

� 78 � 16

� 62 ■

In the next example we evaluate a determinant using expansion by minorsabout the second row. In expanding about any row or column, the signs of thecoefficients of the minors alternate according to the sign array that follows:

� �The sign array is easily remembered by observing that there is a “�” sign in theupper left position and then alternating signs for all of the remaining positions.

� � �

� � �

� � �

3 �54 6

3 �5�7 9

4 6�7 9

1 3 �5�2 4 6

0 �7 9

1 3 �5�2 4 6

0 �7 9

b1 c1b2 c2

b1 c1b3 c3

b2 c2b3 c3

a1 b1 c1a2 b2 c2a3 b3 c3

2 �80 3

2 �80 3

238 (4-40) Chapter 4 Systems of Linear Equations

s t u d y t i p

Remember that everythingwe do in solving problems isbased on principles (which arealso called rules, theorems,and definitions). These princi-ples justify the steps we take.Be sure that you understandthe reasons. If you just memo-rize procedures without un-derstanding, you will soon for-get the procedures.

E X A M P L E 3 Determinant of a 3 � 3 matrixEvaluate the determinant of the matrix by expanding by minors about the secondrow.

� �SolutionFor expansion using the second row we prefix the signs “� � �” from the secondrow of the sign array to the corresponding numbers in the second row of the matrix,�2, 4, and 6. Note that the signs from the sign array are used in addition to anysigns that occur on the numbers in the second row.

From the sign array, second row

� � � �(�2) � � � � 4 � � � � 6 � � �� 2(27 � 35) � 4(9 � 0) � 6(�7 � 0)

� 2(�8) � 4(9) � 6(�7)

� �16 � 36 � 42

� 62

Note that 62 is the same value that was obtained for this determinant in Example 2. ■

It can be shown that expanding by minors using any row or column prefixed bythe corresponding signs from the sign array yields the same value for the determi-nant. Because we can use any row or column to evaluate a determinant of a 3 � 3matrix, we can choose a row or column that makes the work easier. We can shortenthe work considerably by picking a row or column with zeros in it.

E X A M P L E 4 Choosing the simplest row or columnFind the determinant of the matrix

� �SolutionWe choose to expand by minors about the third column of the matrix because thethird column contains two zeros. Prefix the third-column entries 0, 0, 2 by the signs“� � �” from the third column of the sign array:

� � � 0 � � � � 0 � � � � 2 � � �� 0 � 0 � 2[�18 � (�20)]

� 4 ■

3 �54 �6

3 �57 9

4 �67 9

3 �5 04 �6 07 9 2

3 �5 04 �6 07 9 2

1 30 �7

1 �50 9

3 �5�7 9

1 3 �5�2 4 6

0 �7 9

1 3 �5�2 4 6

0 �7 9


A calculator is very useful forfinding the determinant of a3 � 3 matrix. Define A usingMATRX EDIT.

c a l c u l a t o r

c l o s e - u p

Now use the determinantfunction from MATRX MATHand the A from MATRX NAMESto find the determinant.

Cramer’s Rule

A system of three linear equations in three variables can be solved by using deter-minants in a manner similar to that of the previous section. This rule is also calledCramer’s rule.

Cramer’s Rule for Three Equations in Three Unknowns

The solution to the system

a1x � b1y � c1z � d1

a2x � b2y � c2z � d2

a3x � b3y � c3z � d3

is given by x � �DD

x�, y � �

D

Dy

�, and z � �DD

z�, where

D � � �, Dx � � �,

Dy � � �, Dz � � �,provided that D � 0.

Note that Dx , Dy , and Dz are obtained from D by replacing the x-, y-, or z-column with the constants d1, d2, and d3.

E X A M P L E 5 Solving an independent systemUse Cramer’s rule to solve the system:

x � y � z � 4

x � y � �3

x � 2y � z � 0

SolutionWe first calculate D, Dx , Dy , and Dz . To calculate D, expand by minors about thethird column because the third column has a zero in it:

D � � � � 1 � � � � 0 � � � � (�1) � � �� 1 � [2 � (�1)] � 0 � (�1)[�1 � 1]

� 3 � 0 � 2

� 5

1 11 �1

1 11 2

1 �11 2

1 1 11 �1 01 2 �1

a1 b1 d1a2 b2 d2a3 b3 d3

a1 d1 c1a2 d2 c2a3 d3 c3

d1 b1 c1d2 b2 c2d3 b3 c3

a1 b1 c1a2 b2 c2a3 b3 c3


For Dx , expand by minors about the first column:

Dx � � � � 4 � � � � (�3) � � � � 0 � � �� 4 � (1 � 0) � 3 � (�1 � 2) � 0

� 4 � 9 � 0 � �5

For Dy , expand by minors about the third row:

Dy � � � � 1 � � � � 0 � � � � (�1) � � �� 1 � 3 � 0 � (�1)(�7) � 10

To get Dz , expand by minors about the third row:

Dz � � � � 1 � � � � 2 � � � � 0 � � �� 1 � 1 � 2(�7) � 0 � 15

Now, by Cramer’s rule,

x � � � �1, y � � � 2, and z � � � 3.

Check (�1, 2, 3) in the original equations. The solution set is (�1, 2, 3). ■

If D � 0, Cramer’s rule does not apply. Cramer’s rule provides the solutiononly to a system of three equations with three variables that has a single point in thesolution set. If D � 0, then the solution set either is empty or consists of infinitelymany points, and we use elimination of variables to find the solution.

E X A M P L E 6 Solving a dependent systemSolve the system:

(1) x � y � z � 2

(2) 2x � 2y � 2z � 4

(3) �3x � 3y � 3z � �6

SolutionCalculate D by expanding about the first column:

D � � � � 1 � � � � 2 � � � � (�3) � � �� 1 � 0 � 2 � 0 � (�3) � 0 � 0

Because D � 0, Cramer’s rule does not apply to this system. If we multiply Eq. (1)by 2, we get Eq. (2). If we multiply Eq. (1) by �3, we get Eq. (3). Thus all threeequations are equivalent, and they are dependent. The solution set to the system is(x, y, z) x � y � z � 2. ■

1 �12 �2

1 �1�3 3

2 �2�3 3

1 1 �12 2 �2

�3 �3 3

15�5

Dz�D

10�5

Dy�D

�5�5

Dx�D

1 11 �1

1 41 �3

1 4�1 �3

1 1 41 �1 �31 2 0

1 41 �3

1 11 0

4 1�3 0

1 4 11 �3 01 0 �1

1 1�1 0

1 12 �1

�1 02 �1

4 1 1�3 �1 0

0 2 �1


When you see the amount ofarithmetic required to solvethe system in Example 5 byCramer’s rule, you can under-stand why computers andcalculators have been pro-grammed to perform thismethod. Some calculators canfind determinants for matri-ces as large as 10 � 10. Tryto solve Example 5 with agraphing calculator that hasdeterminants.

c a l c u l a t o r

c l o s e - u p

True or false? Explain your answer.

1. A minor is the determinant of a 2 � 2 matrix. True

2. The minor for an element is found by deleting the element from the matrix.False

3. The determinant of a 3 � 3 matrix is found by using minors. True

4. Expansion by minors converts a 3 � 3 matrix into a 4 � 4 matrix. False

5. Using Cramer’s rule, we use �DD

x� to get the value of x. False

6. Expansion by minors about any row or any column gives the same value forthe determinant of a 3 � 3 matrix. True

7. The sign array is used in evaluating the determinant of a 3 � 3 matrix.True

8. It is easier to find the determinant of a 3 � 3 matrix with several zeroelements than one with no zero elements. True

9. If D � 0, then x, y, and z are all zero. False

10. Cramer’s rule solves nonlinear systems of three equations in threeunknowns. False


W A R M - U P S

E X E R C I S E S4 . 6

Reading and Writing After reading this section, write out theanswers to these questions. Use complete sentences.

1. What is a minor?A minor for an element in a 3 � 3 matrix is the determi-nant of a 2 � 2 matrix.

2. How do you find the minor for an element of a 3 � 3matrix?A minor for an element is obtained by deleting the row andcolumn of the element and finding the determinant of the2 � 2 matrix that remains.

3. What is the purpose of the sign array?The sign array tells what signs to use in the expansion byminors.

4. Which systems can be solved by Cramer’s rule for threeequations in three unknowns?Cramer’s rule solves only those systems that have a uniquesolution.

Find the indicated minors using the following matrix. SeeExample 1.

� �5. Minor for 3 11 6. Minor for �2 �24

7. Minor for 5 4 8. Minor for �3 �18

3 �2 54 �3 70 1 �6

9. Minor for 7 3 10. Minor for 0 1

11. Minor for 1 1 12. Minor for �6 �1

Find the determinant of each 3 � 3 matrix by using expansionby minors about the first column. See Example 2.

13. � � �7 14. � � �12 1 31 1 23 4 6

1 1 22 3 13 1 5

15. � � �1 16. � � �51 0 22 1 34 3 0

2 1 01 0 13 1 2

17. � � 9 18. � � �26�2 1 3�1 4 2

2 1 1

�2 1 2�3 3 1�5 4 0

19. � � 5 20. � � 9

Evaluate the determinant of each 3 � 3 matrix using expan-sion by minors about the row or column of your choice. SeeExamples 3 and 4.

21. � � 22 22. � � 32 1 21 2 53 0 0

3 1 52 0 64 0 1

1 0 60 1 40 0 9

1 1 50 3 20 2 3

23. � � 6 24. � � �6

25. � � 70 26. � � �28

27. � � 25 28. � � �1

Use Cramer’s rule to solve each system. See Example 5.

29. x � y � z � 6 30. x � y � z � 2x � y � z � 2 x � y � 2z � �3

2x � y � z � 7 2x � y � z � 7(1, 2, 3) (1, �2, 3)

31. x � 3y � 2z � 0 32. 3x � 2y � 2z � 0x � y � z � 2 x � y � z � 1x � y � z � 0 x � y � z � 3(�1, 1, 2) (2, �1, �2)

33. x � y � �1 34. x � y � 82y � z � 3 x � 2z � 0

x � y � z � 0 x � y � z � 1(�3, 2, 1) (6, �2, 3)

35. x � y � z � 0 36. x � y � z � 12x � 2y � z � 6 5x � y � 0x � 3y � 0 3x � y � 2z � 0

��3

2�, �

1

2�, 2��

1

2�, �

5

2�, �2��

37. x � y � z � 0 38. x � z � 02y � 2z � 0 x � 3y � 1

3x � y � �1 4y � 3z � 3(0, 1, �1) (1, 0, �1)

Solve each system. Use Cramer’s rule if possible. See Exam-ple 6.

39. 2x � y � z � 1 40. x � y � z � 4�6x � 3y � 3z � �3 2x � 2y � 2z � 3

4x � 2y � 2z � 2 4x � y � z � 1(x, y, z) � 2x � y � z � 1

41. x � y � 1 42. x � y � z � 5y � 2z � 3 2x � 2y � 2z � 10

x � 2y � 2z � 5 3x � 3y � 3z � 15 (x, y, z) � x � y � z � 5

43. x � y � 4 44. x � y � 0y � z � �3 x � z � �1

x � z � �5 y � z � 3(1, 3, �6)

2 3 06 4 11 2 0

2 1 10 0 55 0 4

�2 6 30 4 0

�1 �4 5

�2 �3 04 �1 00 3 5

�2 0 1�3 2 �5

4 �2 6

�2 1 30 1 �12 �4 �3


Write a system of three equations in three variables for eachword problem. Use Cramer’s rule to solve each system.

45. Weighing dogs. Cassandra wants to determine the weightsof her two dogs, Mimi and Mitzi. However, neither dog willsit on the scale by herself. Cassandra, Mimi, and Mitzialtogether weigh 175 pounds. Cassandra and Mimi togetherweigh 143 pounds. Cassandra and Mitzi together weigh139 pounds. How much does each weigh individually?Mimi 36 pounds, Mitzi 32 pounds, Cassandra 107 pounds

175

CassandraMimiMitzi

CassandraMimi

CassandraMitzi

143 139

F I G U R E F O R E X E R C I S E 4 5

46. Nickels, dimes, and quarters. Bernard has 41 coins con-sisting of nickels, dimes, and quarters, and they are worth atotal of $4.00. If the number of dimes plus the number ofquarters is one more than the number of nickels, then howmany of each does he have?20 nickels, 15 dimes, 6 quarters

47. Finding three angles. If the two acute angles of a right tri-angle differ by 12°, then what are the measures of the threeangles of this triangle?39°, 51°, 90°

48. Two acute and one obtuse. The obtuse angle of a triangleis twice as large as the sum of the two acute angles. If thesmallest angle is only one-eighth as large as the sum of theother two, then what is the measure of each angle?20°, 40°, 120°

GET TING MORE INVOLVED

49. Writing. For what values of a, b, c, and d is the determi-nant of the matrix

� �equal to zero? Explain your answer.

50. Exploration. The determinant of a 4 � 4 matrix is foundby expanding by 3 � 3 minors and using a 4 � 4 signarray. Find the determinant of a 4 � 4 matrix of yourchoice. Use the definition of the determinant of a 3 � 3matrix as a guide.

a b 0c d 0b a 0

GR APHING CALCULATOREXERCISES

51. Use the determinant feature of your graphing calculator tofind the determinant of each matrix in Exercises 13–20 ofthis section.

52. Solve the systems in Exercises 29–38 by using Cramer’srule and your graphing calculator to find the determinants.


53. The solution to an independent system of four linear equa-tions in four variables can be found by using determinantsof 4 � 4 matrices in the same manner as Cramer’s rule forthree variables. Write Cramer’s rule for four variables.Make up a system of four linear equations in four variablesand solve it using your new rule and a graphing calculatorto evaluate the determinants.

L I N E A R P R O G R A M M I N G

In this section we graph the solution set to a system of several linear inequalities,much as we graphed compound linear inequalities in Chapter 3. We then use thesolution set as the domain of a function for which we are seeking the maximum orminimum value. The method that we use is called linear programming, and itcan be applied to problems such as finding maximum profit or minimum cost.

Graphing the Constraints

In linear programming we have two variables that must satisfy several linearinequalities. These inequalities are called the constraints because they restrict thevariables to only certain values. A graph in the coordinate plane is used to indicatethe points that satisfy all of the constraints.

E X A M P L E 1 Graphing the constraintsGraph the solution set to the system of inequalities and identify each vertex of theregion:

x 0, y 0

3x � 2y � 12

x � 2y � 8

SolutionThe points on or to the right of the y-axis satisfy x 0. The points on or above thex-axis satisfy y 0. The points on or below the line 3x � 2y � 12 satisfy3x � 2y � 12. The points on or below the line x � 2y � 8 satisfy x � 2y � 8.Graph each straight line and shade the region that satisfies all four inequalities asshown in Fig. 4.7. Three of the vertices are easily identified as (0, 0), (0, 4),and (4, 0). The fourth vertex is found by solving the system 3x � 2y � 12 andx � 2y � 8. The fourth vertex is (2, 3). ■

In linear programming the constraints usually come from physical limitations insome problem. In the next example we write the constraints and then graph thepoints in the coordinate plane that satisfy all of the constraints.

E X A M P L E 2 Writing the constraintsJules is in the business of constructing dog houses. A small dog house requires8 square feet (ft2) of plywood and 6 ft2 of insulation. A large dog house requires16 ft2 of plywood and 3 ft2 of insulation. Jules has available only 48 ft2 of plywood

4.7

I n t h i ss e c t i o n

● Graphing the Constraints

● Maximizing or Minimizing aLinear Function

y

x

–2

1

–1–1

2

3

31

2

5(0, 4)

(2, 3)

(4, 0)

x � 2y � 8

3x � 2y � 12

F I G U R E 4 . 7

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