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SECTION 5.5
1
Solving Systems of Equations by Using Determinants
SECTION
5.5
OBJECTIVE A
Point of Interest
The word matrix was first
used in a mathematical
context in 1850. The root of
this word is the Latin mater,
meaning mother. A matrix
was thought of as an object
from which something else
originates. The idea was that
a determinant, discussed
below, originated (was born)
from a matrix. Today, matrices
are one of the most widely
used tools of applied
mathematics.
Solving Systems of Equations by
Using Determinants
To evaluate a determinant
A matrix is a rectangular array of numbers. Each
number in a matrix is called an element of the matrix.
The matrix at the right, with three rows and four
columns, is called a 3 ⫻ 4 (read “3 by 4”) matrix.
册
冋 册
冋 册
A matrix of m rows and n columns is said to be of order m ⫻ n. The matrix above has
order 3 ⫻ 4. The notation aij refers to the element of a matrix in the ith row and the jth
column. For matrix A, a23 苷 ⫺3, a31 苷 6, and a13 苷 2.
A square matrix is one that has the same number of
rows as columns. A 2 ⫻ 2 matrix and a 3 ⫻ 3 matrix
are shown at the right. Associated with every square
matrix is a number called its determinant.
Determinant of a 2 ⫻ 2 Matrix
The determinant of a 2 ⫻ 2 matrix
Take Note
冋
1 ⫺3 2 4
A 苷 0 4 ⫺3 2
6 ⫺5 4 ⫺1
is given by the formula
冟
冟
4 0 1
5 ⫺3 7
2 1 4
冟
a 11 a 12
a a
is written 11 12 . The value of this determinant
a 21 a 22
a 21 a 22
冟
Note that vertical bars are
used to represent the
determinant and that brackets
are used to represent the
matrix.
HOW TO • 1
冋 册
⫺1 3
5 2
冟
a 11 a 12
苷 a 11 a 22 ⫺ a 12 a 21
a 21 a 22
Find the value of the determinant
冟
冟
冟
3 4
.
⫺1 2
3 4
苷 3 ⭈ 2 ⫺ 4共⫺1兲 苷 6 ⫺ 共⫺4兲 苷 10
⫺1 2
The value of the determinant is 10.
For a square matrix whose order is 3 ⫻ 3 or greater, the value of the determinant is found
by using 2 ⫻ 2 determinants.
The minor of an element in a 3 ⫻ 3 determinant is the 2 ⫻ 2 determinant that is
obtained by eliminating the row and column that contain that element.
HOW TO • 2
Find the minor of ⫺3 for the determinant
ⱍ ⱍ
2 ⫺3 4
0 4 8.
⫺1 3 6
The minor of ⫺3 is the 2 ⫻ 2 determinant created by eliminating the row and
column that contain ⫺3.
ⱍ ⱍ
2 ⫺3 4
Eliminate the row and column as shown: 0 4 8
⫺1 3 6
0 8
The minor of ⫺3 is
.
⫺1 6
冟
冟
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Systems of Linear Equations and Inequalities
Take Note
The only difference between
the cofactor and the minor of
an element is one of sign.
The definition at the right can
be stated with the use of
symbols as follows: If C i j is
the cofactor and M i j is the
minor of the matrix element
ai j , then C i j 苷 共⫺1兲i ⫹j M i j . If
i ⫹ j is an even number, then
共⫺1兲i ⫹j 苷 1 and C i j 苷 M i j . If
i ⫹ j is an odd number, then
共⫺1兲i ⫹j 苷 ⫺1 and C i j 苷 ⫺M i j .
Cofactor of an Element of a Matrix
The cofactor of an element of a matrix is 共⫺1兲i ⫹ j times the minor of that element, where i is the
row number of the element and j is the column number of the element.
HOW TO • 3
of ⫺5.
ⱍ ⱍ
3 ⫺2 1
For the determinant 2 ⫺5 ⫺4 , find the cofactor of ⫺2 and
0 3 1
Because ⫺2 is in the first row and the second column, i 苷 1 and j 苷 2. Thus
2 ⫺4
.
共⫺1兲i⫹j 苷 共⫺1兲1⫹2 苷 共⫺1兲3 苷 ⫺1. The cofactor of ⫺2 is 共⫺1兲
0 1
冟
冟
Because ⫺5 is in the second row and the second column, i 苷 2 and j 苷 2.
Thus 共⫺1兲i⫹j 苷 共⫺1兲2⫹2 苷 共⫺1兲4 苷 1. The cofactor of ⫺5 is 1 ⭈
冟 冟
3 1
.
0 1
Note from this example that the cofactor of an element is ⫺1 times the minor of that
element or 1 times the minor of that element, depending on whether the sum i ⫹ j is an
odd or an even integer.
The value of a 3 ⫻ 3 or larger determinant can be found by expanding by cofactors of
any row or any column.
HOW TO • 4
ⱍ ⱍ
ⱍ ⱍ
2 ⫺3 2
Find the value of the determinant 1 3 ⫺1 .
0 ⫺2 2
We will expand by cofactors of the first row. Any row or column would work.
2 ⫺3 2
3 ⫺1
1 ⫺1
1 3
1 3 ⫺1 苷 2(⫺1)1⫹1
⫹ 共⫺3兲(⫺1)1⫹2
⫹ 2(⫺1)1⫹3
⫺2 2
0 2
0 ⫺2
0 ⫺2 2
3 ⫺1
1 ⫺1
1 3
⫹ 共⫺3兲(⫺1)
⫹ 2(1)
苷 2(1)
⫺2 2
0 2
0 ⫺2
冟
冟
冟
冟
冟
冟
冟
冟
冟
冟
冟
冟
⫽ 2(6 ⫺ 2) ⫹ 3(2 ⫺ 0) ⫹ 2(⫺2 ⫺ 0)
⫽ 2(4) ⫹ 3(2) ⫹ 2(⫺2) ⫽ 8 ⫹ 6 ⫺ 4 ⫽ 10
To illustrate that any row or column can be chosen when expanding by cofactors, we will
now show the evaluation of the same determinant by expanding by cofactors of the second column.
ⱍ ⱍ
2 ⫺3 2
1 ⫺1
2 2
2 2
1 3 ⫺1 苷 ⫺3共⫺1兲1⫹2
⫹ 3共⫺1兲2⫹2
⫹ 共⫺2兲共⫺1兲3⫹2
0 2
0 2
1 ⫺1
0 ⫺2 2
冟
苷 ⫺3共⫺1兲
冟
冟
冟
冟 冟
冟 冟
冟
冟
1 ⫺1
2 2
2 2
⫹ 3共1兲
⫹ 共⫺2兲共⫺1兲
0 2
0 2
1 ⫺1
苷 3共2 ⫺ 0兲 ⫹ 3共4 ⫺ 0兲 ⫹ 2共⫺2 ⫺ 2兲
苷 3共2兲 ⫹ 3共4兲 ⫹ 2共⫺4兲 苷 6 ⫹ 12 ⫹ 共⫺8兲 苷 10
冟
冟
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SECTION 5.5
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Solving Systems of Equations by Using Determinants
3
Note that the value of the determinant is the same whether the first row or the second
column is used to expand by cofactors. Any row or column can be used to evaluate a
determinant by expanding by cofactors.
EXAMPLE • 1
Find the value of
YOU TRY IT • 1
冟
冟
3 ⫺2
6 ⫺4 .
Find the value of
Solution
3 ⫺2
苷 3共⫺4兲 ⫺ 共⫺2兲共6兲 苷 ⫺12 ⫹ 12 苷 0
6 ⫺4
冟
冟
冟
冟
⫺1 ⫺4
.
3 ⫺5
Your solution
The value of the determinant is 0.
EXAMPLE • 2
ⱍ ⱍ
YOU TRY IT • 2
ⱍ ⱍ
⫺2 3 1
Find the value of 4 ⫺2 0 .
1 ⫺2 3
1 4 ⫺2
Find the value of 3 1 1 .
0 ⫺2 2
Solution
Expand by cofactors of the first row.
Your solution
ⱍ 冟 ⱍ冟
⫺2 3 1
4 ⫺2 0
1 ⫺2 3
⫺2 0
4 0
4 ⫺2
苷 ⫺2
⫺3
⫹1
⫺2 3
1 3
1 ⫺2
苷 ⫺2共⫺6 ⫺ 0兲 ⫺ 3共12 ⫺ 0兲 ⫹ 1共⫺8 ⫹ 2兲
苷 ⫺2共⫺6兲 ⫺ 3共12兲 ⫹ 1共⫺6兲
苷 12 ⫺ 36 ⫺ 6
苷 ⫺30
冟 冟 冟
冟
The value of the determinant is ⫺30.
EXAMPLE • 3
ⱍ ⱍ
YOU TRY IT • 3
0 ⫺2 1
Find the value of 1 4 1 .
2 ⫺3 4
ⱍ 冟 ⱍ冟
Solution
0 ⫺2 1
1 4 1
2 ⫺3 4
冟 冟 冟
4 1
1 1
1 4
⫺ 共⫺2兲
⫹1
⫺3 4
2 4
2 ⫺3
苷 0 ⫺ 共⫺2兲共4 ⫺ 2兲 ⫹ 1共⫺3 ⫺ 8兲
苷 2共2兲 ⫹ 1共⫺11兲
苷 4 ⫺ 11
苷 ⫺7
苷0
The value of the determinant is ⫺7.
ⱍ ⱍ
3 ⫺2 0
Find the value of
1 4 2.
⫺2 1 3
Your solution
冟
Solutions on p. S1
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Systems of Linear Equations and Inequalities
OBJECTIVE B
To solve a system of equations by using Cramer’s Rule
The connection between determinants and systems of equations can be understood by
solving a general system of linear equations.
Solve: (1) a1 x ⫹ b1 y 苷 c1
(2) a2 x ⫹ b2 y 苷 c2
Eliminate y. Multiply Equation (1) by b2 and Equation (2) by ⫺b1.
a1 b2 x ⫹ b1 b2 y 苷 c1 b2
⫺a2 b1 x ⫺ b1 b2 y 苷 ⫺c2 b1
• b2 times Equation (1)
• ⫺b1 times Equation (2)
Add the equations.
a1 b2 x ⫺ a2 b1 x 苷 c1 b2 ⫺ c2 b1
共a1 b2 ⫺ a2 b1兲x 苷 c1 b2 ⫺ c2 b1
c b ⫺ c2 b1
x苷 1 2
a1 b2 ⫺ a2 b1
• Solve for x, assuming a1b2 ⫺ a2b1 ⫽ 0.
The denominator a1 b2 ⫺ a2 b1 is the determinant of the coefficients of x and y. This is
called the coefficient determinant.
a1 b2 ⫺ a2 b1 苷
coefficients of x
coefficients of y
The numerator c1 b2 ⫺ c2 b1 is the determinant
obtained by replacing the first column in
the coefficient determinant by the constants
c1 and c2. This is called a numerator determinant.
Point of Interest
Cramer’s Rule is named after
Gabriel Cramer, who used it
in a book he published in
1750. However, this rule was
also published in 1683 by the
Japanese mathematician Seki
Kowa. That publication
occurred seven years before
Cramer’s birth.
c1 b2 ⫺ c2 b1 苷
冟
a1 b1
a2 b2
冟
冟 冟
c1 b1
c2 b2
constants of
the equations
By following a similar procedure and eliminating x, it is also possible to express the ycomponent of the solution in determinant form. These results are summarized in
Cramer’s Rule.
Cramer’s Rule
The solution of the system of equations
where D 苷
冟
冟
冟
Solve by using Cramer’s Rule:
冟
3 ⫺2
苷 19
2 5
冟
冟 冟
a1 b1
c1 b 1
a 1 c1
, Dx 苷
, Dy 苷
, and D ⫽ 0.
a2 b2
c2 b 2
a 2 c2
HOW TO • 5
D苷
冟 冟
D
D
a 1 x ⫹ b 1 y 苷 c1
is given by x 苷 x and y 苷 y ,
a 2 x ⫹ b 2 y 苷 c2
D
D
3x ⫺ 2y 苷 1
2x ⫹ 5y 苷 3
• Find the value of the coefficient determinant.
冟
冟 冟
1 ⫺2
3 1
苷 11, Dy 苷
苷 7 • Find the value of each of the
3 5
2 3
numerator determinants.
11
7
Dx
Dy
x苷
苷 ,y苷
苷
• Use Cramer’s Rule to write the solution.
D
19
D
19
Dx 苷
The solution is
冉 冊
11 7
,
19 19
.
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SECTION 5.5
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Solving Systems of Equations by Using Determinants
5
A procedure similar to that followed for two equations in two variables can be used to
extend Cramer’s Rule to three equations in three variables.
Cramer’s Rule for a System of Three Equations in Three Variables
The solution of the system of equations
is given by x 苷
a 1 x ⫹ b 1 y ⫹ c 1z 苷 d 1
a 2 x ⫹ b 2 y ⫹ c 2z 苷 d 2
a 3 x ⫹ b 3 y ⫹ c 3z 苷 d 3
Dx
Dy
Dz
, y 苷 , and z 苷 , where
D
D
D
ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ
a 1 b 1 c1
d 1 b 1 c1
a 1 d 1 c1
a1 b1 d1
D 苷 a 2 b 2 c2 , Dx 苷 d 2 b 2 c2 , Dy 苷 a 2 d 2 c2 , Dz 苷 a 2 b 2 d 2 , and D ⫽ 0.
a 3 b 3 c3
d 3 b 3 c3
a 3 d 3 c3
a3 b3 d3
HOW TO • 6
Solve by using Cramer’s Rule:
ⱍ ⱍ冟 冟
ⱍ ⱍ冟 冟
冟 冟
ⱍ ⱍ
冟 冟
ⱍ ⱍ
2x ⫺ y ⫹ z 苷 1
x ⫹ 3y ⫺ 2z 苷 ⫺2
3x ⫹ y ⫹ 3z 苷 4
Find the value of the coefficient determinant.
2 ⫺1 1
3 ⫺2
1 ⫺2
1 3
D 苷 1 3 ⫺2 苷 2
⫺ 共⫺1兲
⫹1
1 3
3 3
3 1
3 1 3
苷 2共11兲 ⫹ 1共9兲 ⫹ 1共⫺8兲
苷 23
冟
冟 冟 冟
Find the value of each of the numerator determinants.
1 ⫺1 1
3 ⫺2
⫺2 ⫺2
⫺2 3
Dx 苷 ⫺2 3 ⫺2 苷 1
⫺ 共⫺1兲
⫹1
1 3
4 3
4 1
4 1 3
苷 1共11兲 ⫹ 1共2兲 ⫹ 1共⫺14兲
苷 ⫺1
冟
冟
冟 冟
冟 冟
2 1 1
⫺2 ⫺2
1 ⫺2
1 ⫺2
Dy 苷 1 ⫺2 ⫺2 苷 2
⫺1
⫹1
4 3
3 3
3 4
3 4 3
苷 2共2兲 ⫺ 1共9兲 ⫹ 1共10兲
苷5
2 ⫺1 1
3 ⫺2
1 ⫺2
1 3
Dz 苷 1 3 ⫺2 苷 2
⫺ 共⫺1兲
⫹1
1 4
3 4
3 1
3 1 4
苷 2共14兲 ⫹ 1共10兲 ⫹ 1共⫺8兲
苷 30
冟
冟 冟 冟
Use Cramer’s Rule to write the solution.
⫺1
D
5
D
30
D
,
y苷 y苷
,
z苷 z苷
x苷 x苷
D
23
D
23
D
23
冉
The solution is ⫺
1 5 30
, ,
23 23 23
冊
.
冟
冟
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Systems of Linear Equations and Inequalities
EXAMPLE • 4
YOU TRY IT • 4
Solve by using Cramer’s Rule.
6x ⫺ 9y 苷 5
4x ⫺ 6y 苷 4
Solve by using Cramer’s Rule.
3x ⫺ y 苷 4
6x ⫺ 2y 苷 5
Solution
6 ⫺9
D苷
苷0
4 ⫺6
Your solution
冟
冟
Dx
is undefined. Therefore, the
D
system is dependent or inconsistent.
Because D 苷 0,
EXAMPLE • 5
YOU TRY IT • 5
Solve by using Cramer’s Rule.
3x ⫺ y ⫹ z 苷 5
x ⫹ 2y ⫺ 2z 苷 ⫺3
2x ⫹ 3y ⫹ z 苷 4
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
Solution
3 ⫺1 1
D 苷 1 2 ⫺2 苷 28,
2 3 1
Solve by using Cramer’s Rule.
2x ⫺ y ⫹ z 苷 ⫺1
3x ⫹ 2y ⫺ z 苷 3
x ⫹ 3y ⫹ z 苷 ⫺2
Your solution
ⱍ
5 ⫺1 1
Dx 苷 ⫺3 2 ⫺2 苷 28,
4 3 1
ⱍ
ⱍ
3 5 1
Dy 苷 1 ⫺3 ⫺2 苷 0,
2 4 1
3 ⫺1 5
Dz 苷 1 2 ⫺3 苷 56
2 3 4
Dx
28
苷
苷1
D
28
D
0
y苷 y苷
苷0
D
28
D
56
z苷 z苷
苷2
D
28
x苷
The solution is (1, 0, 2).
Solutions on p. S1
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SECTION 5.5
•
Solving Systems of Equations by Using Determinants
7
5.5 EXERCISES
OBJECTIVE A
To evaluate a determinant
1. How do you find the value of the determinant associated with a 2 ⫻ 2
matrix?
2. What is the cofactor of a given element in a matrix?
For Exercises 3 to 14, evaluate the determinant.
3.
冟
7.
冟 冟
11.
2 ⫺1
3 4
冟
3 6
2 4
ⱍ ⱍ
3 ⫺1 2
0 1 2
3 2 ⫺2
4.
冟
5 1
⫺1 2
8.
冟
5 ⫺10
1 ⫺2
12.
冟
冟
5.
冟
ⱍ ⱍ
4 5 ⫺2
3 ⫺1 5
2 1 4
6 ⫺2
⫺3 4
冟
ⱍ ⱍ
ⱍ ⱍ
1 ⫺1 2
3 2 1
1 0 4
9.
4 2 6
⫺2 1 1
2 1 3
13.
6.
10.
14.
冟
⫺3 5
1 7
冟
ⱍ ⱍ
ⱍ ⱍ
4 1 3
2 ⫺2 1
3 1 2
3 6 ⫺3
4 ⫺1 6
⫺1 ⫺2 3
15. What is the value of a determinant for which one row is all zeros?
16. What is the value of a determinant for which all the elements are the same number?
OBJECTIVE B
To solve a system of equations by using Cramer’s Rule
For Exercises 17 to 34, solve by using Cramer’s Rule.
17.
2x ⫺ 5y 苷 26
5x ⫹ 3y 苷 3
18.
3x ⫹ 7y 苷 15
2x ⫹ 5y 苷 11
19.
x ⫺ 4y 苷 8
3x ⫹ 7y 苷 5
20. 5x ⫹ 2y 苷 ⫺5
3x ⫹ 4y 苷 11
21.
2x ⫹ 3y 苷 4
6x ⫺ 12y 苷 ⫺5
22.
5x ⫹ 4y 苷 3
15x ⫺ 8y 苷 ⫺21
23.
2x ⫹ 5y 苷 6
6x ⫺ 2y 苷 1
24. 7x ⫹ 3y 苷 4
5x ⫺ 4y 苷 9
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Systems of Linear Equations and Inequalities
25.
⫺2x ⫹ 3y 苷 7
4x ⫺ 6y 苷 9
26. 9x ⫹ 6y 苷 7
3x ⫹ 2y 苷 4
29.
2x ⫺ y ⫹ 3z 苷 9
x ⫹ 4y ⫹ 4z 苷 5
3x ⫹ 2y ⫹ 2z 苷 5
30.
32.
x ⫹ 2y ⫹ 3z 苷 8
2x ⫺ 3y ⫹ z 苷 5
3x ⫺ 4y ⫹ 2z 苷 9
33. 4x ⫺ 2y ⫹ 6z 苷 1
3x ⫹ 4y ⫹ 2z 苷 1
2x ⫺ y ⫹ 3z 苷 2
27.
2x ⫺ 5y 苷 ⫺2
3x ⫺ 7y 苷 ⫺3
3x ⫺ 2y ⫹ z 苷 2
2x ⫹ 3y ⫹ 2z 苷 ⫺6
3x ⫺ y ⫹ z 苷 0
28.
31.
34.
8x ⫹ 7y 苷 ⫺3
2x ⫹ 2y 苷 5
3x ⫺ y ⫹ z 苷 11
x ⫹ 4y ⫺ 2z 苷 ⫺12
2x ⫹ 2y ⫺ z 苷 ⫺3
x ⫺ 3y ⫹ 2z 苷 1
2x ⫹ y ⫺ 2z 苷 3
3x ⫺ 9y ⫹ 6z 苷 ⫺3
35. Can Cramer’s Rule be used to solve a dependent system of equations?
36. Suppose a system of linear equations in two variables has Dx ⫽ 0, Dy ⫽ 0, and D ⫽ 0. Is the system of equations independent, dependent, or inconsistent?
Applying the Concepts
37. Determine whether the following statements are always true, sometimes true, or
never true.
a. The determinant of a matrix is a positive number.
b. A determinant can be evaluated by expanding about any row or column of the
matrix.
c. Cramer’s Rule can be used to solve a system of linear equations in three variables.
38. Show that
冟 冟
冟 冟
a b
c d
苷⫺
.
c d
a b
A苷
冦冟
冟 冟 冟 冟 冟
冟 冟冧
1 x1 x2
x x
x x
x x
⫹ 2 3 ⫹ 3 4 ⫹ ⭈ ⭈ ⭈ ⫹ n 1
2 y1 y2
y2 y3
y3 y4
yn y1
Use the surveyor’s area formula to find the area of the polygon with vertices
共9, ⫺3兲, (26, 6), (18, 21), (16, 10), and (1, 11). Measurements are given in feet.
© Spencer Grant/PhotoEdit, Inc.
39. Surveying Surveyors use a formula to find the area of a plot of land. The surveyor’s area formula states that if the vertices 共x1, y1兲, 共x2, y2兲, . . .,共xn, yn兲 of a simple
polygon are listed counterclockwise around the perimeter, then the area of the polygon is
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Solutions to You Try It
You Try It 5
SOLUTIONS TO CHAPTER 5 “YOU TRY IT”
SECTION 5.5
You Try It 1
冟
冟
⫺1 ⫺4
苷 ⫺1共⫺5兲 ⫺ 共⫺4兲共3兲
3 ⫺5
苷 5 ⫹ 12 苷 17
The value of the determinant is 17.
ⱍ 冟ⱍ冟
You Try It 2 Expand by cofactors of the first row.
1 4 ⫺2
3 1 1
0 ⫺2 2
冟 冟
冟
1 1
3 1
3 1
⫺4
⫹ 共⫺2兲
⫺2 2
0 2
0 ⫺2
苷 1共2 ⫹ 2兲 ⫺ 4共6 ⫺ 0兲 ⫺ 2共⫺6 ⫺ 0兲
苷 4 ⫺ 24 ⫹ 12
苷 ⫺8
苷1
冟
The value of the determinant is ⫺8.
You Try It 3
ⱍ 冟 ⱍ冟
3 ⫺2 0
1 4 2
⫺2 1 3
冟
冟 冟
4 2
1 2
1 4
⫺ 共⫺2兲
⫹0
苷3
1 3
⫺2 3
⫺2 1
苷 3共12 ⫺ 2兲 ⫹ 2共3 ⫹ 4兲 ⫹ 0
苷 3共10兲 ⫹ 2共7兲
苷 30 ⫹ 14
苷 44
The value of the determinant is 44.
You Try It 4
3x ⫺ y 苷 4
6x ⫺ 2y 苷 5
3 ⫺1
苷0
D苷
6 ⫺2
冟
冟
Because D 苷 0,
Dx
D
is undefined.
Therefore, the system of equations is
dependent or inconsistent. That is, the
system is not independent.
冟
2 x ⫺ y ⫹ z 苷 ⫺1
3x ⫹ 2y ⫺ z 苷 3
x ⫹ 3y ⫹ z 苷 ⫺2
2 ⫺1 1
D 苷 3 2 ⫺1 苷 21
1 3 1
⫺1 ⫺1 1
Dx 苷 3 2 ⫺1 苷 9
⫺2 3 1
2 ⫺1 1
Dy 苷 3 3 ⫺1 苷 ⫺3
1 ⫺2 1
2 ⫺1 ⫺1
Dz 苷 3 2 3 苷 ⫺42
1 3 ⫺2
9
3
Dx
苷
苷
x苷
D
21
7
⫺3
1
Dy
苷
苷⫺
y苷
D
21
7
⫺42
Dz
苷
苷 ⫺2
z苷
D
21
ⱍ
ⱍ
ⱍ
ⱍ
The solution is
ⱍ
ⱍ
ⱍ
ⱍ
冉
3
1
,⫺ ,
7
7
冊
⫺2 .
S1
46951_20_5.5online.qxd
1/13/10
10:58 AM
Page A1
Answers to Selected Exercises
A1
ANSWERS TO CHAPTER 5 SELECTED EXERCISES
SECTION 5.5
3. 11
5. 18
7. 0
25. Not independent
true
b. Always true
9. 15
27. 共⫺1, 0兲
11. ⫺30
13. 0
29. 共1, ⫺1, 2兲
c. Sometimes true
17. 共3, ⫺4兲
15. 0
31. 共2, ⫺2, 3兲
2
39. The area is 239 ft .
19. 共4, ⫺1兲
33. Not independent
21.
冉, 冊
35. No
11 17
14 21
23.
冉 , 1冊
1
2
37. a. Sometimes