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S Y S T E M S O F L I N E A R E Q U A T I O N S
I N T H R E E V A R I A B L E S
The techniques that you learned in Section 4.2 can be extended to systems of equa-
tions in more than two variables. In this section we use elimination of variables to
solve systems of equations in three variables.
Definition
The equation 5 x 4 y 7 is called a linear equation in two variables because its
graph is a straight line. The equation 2 x 3 y 4 z 12 is similar in form, and so
it is a linear equation in three variables. An equation in three variables is graphed in
a three-dimensional coordinate system. The graph of a linear equation in three vari-
ables is a plane, not a line. We will not graph equations in three variables in this text,
but we can solve systems without graphing. In general, we make the following
definition.
Linear Equation in Three Variables
If A , B , C , and D are real numbers, with A , B , and C not all zero, then
Ax By Cz D
is called a linear equation in three variables.
Solving a System by Elimination
A solution to an equation in three variables is an ordered triple such as
(2, 1, 5), where the first coordinate is the value of x , the second coordinate is the
value of y , and the third coordinate is the value of z. There are infinitely many
solutions to a linear equation in three variables.
The solution to a system of equations in three variables is the set of all ordered
triples that satisfy all of the equations of the system. The techniques for solving a
system of linear equations in three variables are similar to those used on systems of
linear equations in two variables. We eliminate variables by either substitution or
addition.
E X A M P L E 1 A linear system with a single solution
Solve the system:
(1) x y z 1
(2) 2 x 2 y 3 z 8
(3) 2 x y 2 z 9
Solution
We can eliminate z from Eqs. (1) and (2) by multiplying Eq. (1) by 3 and adding it
to Eq. (2):
3 x 3 y 3 z 3 Eq. (1) multiplied by 3
2 x 2 y 3 z 8 Eq. (2)
(4) 5 x y 5
4.3 Systems of Linear Equations in Three Variables (4-19) 217
I n t h i s
s e c t i o n
(^) Definition (^) Solving a System by Elimination (^) Graphs of Equations in Three Variables (^) Applications
s t u d y t i p
Everyone knows that you must practice to be successful with musical instruments, foreign languages, and sports. Success in algebra also re- quires regular practice. Thus budget your time so that you have a regular practice period for algebra.
Now we must eliminate the same variable, z , from another pair of equations. Eliminate z from (1) and (3):
2 x 2 y 2 z 2 Eq. (1) multiplied by 2 2 x y 2 z 9 Eq. (3) (5) 4 x y 7
Equations (4) and (5) give us a system with two variables. We now solve this system. Eliminate y by multiplying Eq. (5) by 1 and adding the equations:
5 x y 5 Eq. (4) 4 x y 7 Eq. (5) multiplied by 1 x 2
Now that we have x , we can replace x by 2 in Eq. (5) to find y :
4 x y 7 4(2) y 7 8 y 7 y 15
Now replace x by 2 and y by 15 in Eq. (1) to find z :
x y z 1 2 15 z 1 13 z 1 z 14 z 14
Check that (2, 15, 14) satisfies all three of the original equations. The solution set is (2, 15, 14).
The strategy that we follow for solving a system of three linear equations in three variables is stated as follows.
218 (4-20) Chapter 4 Systems of Linear Equations
1. Use substitution or addition to eliminate any one of the variables from a pair of equations of the system. Look for the easiest variable to eliminate. 2. Eliminate the same variable from another pair of equations of the system. 3. Solve the resulting system of two equations in two unknowns. 4. After you have found the values of two of the variables, substitute into one of the original equations to find the value of the third variable. 5. Check the three values in all of the original equations.
Solving a System in Three Variables
You can use a calculator to check that (2, 15, 14) satisfies all three equations of the orig- inal system.
c a l c u l a t o r
c l o s e - u p
h e l p f u l h i n t
Note that we could have cho- sen to eliminate x, y, or z first in Example 1.You should solve this same system by eliminat- ing x first and then by elimi- nating y first. To eliminate x first, multiply the first equation by 2 and add it with the sec- ond and third equations.
In the next example we use a combination of addition and substitution.
E X A M P L E 2 Using addition and substitution
Solve the system: (1) x y 4 (2) 2 x 3 z 14 (3) 2 y z 2
E X A M P L E 3 An inconsistent system of three linear equations
Solve the system: (1) x y z 5 (2) 3 x 2 y z 8 (3) 2 x 2 y 2 z 7
Solution We can eliminate the variable z from Eqs. (1) and (2) by adding them: (1) x y z 5 (2) 3 x 2 y z 8 4 x y 13 To eliminate z from Eqs. (1) and (3), multiply Eq. (1) by 2 and add the resulting equation to Eq. (3): 2 x 2 y 2 z 10 Eq. (1) multiplied by 2 2 x 2 y 2 z 7 Eq. (3) 0 3 Because the last equation is false, there are two inconsistent equations in the system. Therefore the solution set is the empty set.
E X A M P L E 4 A dependent system of three equations
Solve the system: (1) 2 x 3 y z 4 (2) 6 x 9 y 3 z 12 (3) 4 x 6 y 2 z 8
Solution We will first eliminate x from Eqs. (1) and (2). Multiply Eq. (1) by 3 and add the resulting equation to Eq. (2): 6 x 9 y 3 z 12 Eq. (1) multiplied by 3 6 x 9 y 3 z 12 Eq. (2) 0 0
have either three different planes intersecting along a line, as in Fig. 4.6(d) or two equations for the same plane, with the third plane intersecting that plane. If all three equations are equations of the same plane, we get that plane for the intersection. We will not solve systems corresponding to all of the possible configurations described. The following examples illustrate two of these cases.
220 (4-22) Chapter 4 Systems of Linear Equations
h e l p f u l h i n t
If you recognize that multiply- ing Eq. (1) by 3 will produce Eq. (2), and multiplying Eq. (1) by 2 will produce Eq. (3), then you can conclude that all three equations are equiva- lent and there is no need to add the equations.
(a) (b) (c) (d)
F I G U R E 4. 6
The last statement is an identity. The identity occurred because Eq. (2) is a multiple of Eq. (1). In fact, Eq. (3) is also a multiple of Eq. (1). These equations are depen- dent. They are all equations for the same plane. The solution set is the set of all points on that plane,
( x , y , z ) 2 x 3 y z 4 .
Applications
Problems involving three unknown quantities can often be solved by using a system of three equations in three variables.
E X A M P L E 5 Finding three unknown rents
Theresa took in a total of $1,240 last week from the rental of three condominiums. She had to pay 10% of the rent from the one-bedroom condo for repairs, 20% of the rent from the two-bedroom condo for repairs, and 30% of the rent from the three- bedroom condo for repairs. If the three-bedroom condo rents for twice as much as the one-bedroom condo and her total repair bill was $276, then what is the rent for each condo?
Solution Let x , y , and z represent the rent on the one-bedroom, two-bedroom, and three- bedroom condos, respectively. We can write one equation for the total rent, another equation for the total repairs, and a third equation expressing the fact that the rent for the three-bedroom condo is twice that for the one-bedroom condo:
x y z 1240 0.1 x 0.2 y 0.3 z 276 z 2 x
Substitute z 2x into both of the other equations to eliminate z :
x y 2 x 1240 0.1 x 0.2 y 0.3(2 x ) 276
3 x y 1240 0.7 x 0.2 y 276
2(3 x y ) 2(1240) Multiply each side by 2. 10(0.7 x 0.2 y ) 10(276) Multiply each side by 10.
6 x 2 y 2480 7 x 2 y 2760 Add. x 280
z 2(280) 560 Because z 2 x 280 y 560 1240 Because x y z 1240 y 400
Check that (280, 400, 560) satisfies all three of the original equations. The condos rent for $280, $400, and $560 per week.
4.3 Systems of Linear Equations in Three Variables (4-23) 221
h e l p f u l (^) h i n t
A problem involving two un- knowns can often be solved with one variable as in Chap- ter 2. Likewise, you can often solve a problem with three unknowns using only two variables. Solve Example 5 by letting a, b, and 2 a be the rent for a one-bedroom, two-bedroom, and a three- bedroom condo.
19. x y 7 20. 2 x y 8 y z 1 y 3 z 22 x 3 z 18 x z 8 (3, 4, 5) (2, 4, 6)
Solve each system. See Examples 3 and 4.
21. x y 2 z 3 2 x y z 5 3 x 3 y 6 z 4 22. 2 x 4 y 6 z 12 6 x 12 y 18 z 36 x 2 y 3 z 6 ( x , y , z ) x 2 y 3 z 6 23. 3 x y z 5 9 x 3 y 3 z 15 12 x 4 y 4 z 20 ( x , y , z ) 3 x y z 5 24. 4 x 2 y 2 z 5 2 x y z 7 4 x 2 y 2 z 6 25. x y 3 y z 8 2 x 2 z 7 26. 2 x y 6 2 y z 4 8 x 2 z 3 27. 0.10 x 0.08 y 0.04 z 3 5 x 4 y 2 z 150 0.3 x 0.24 y 0.12 z 9 ( x , y , z ) 5 x 4 y 2 z 150 28. 0.06 x 0.04 y z 6 3 x 2 y 50 z 300 0.03 x 0.02 y 0.5 z 3 ( x , y , z ) 3 x 2 y 50 z 300
Use a calculator to solve each system.
29. 3 x 2 y 0.4 z 0. 3.7 x 0.2 y 0.05 z 0. 2 x 3.8 y 2.1 z 3.26 (0.1, 0.3, 2) 30. 3 x 0.4 y 9 z 1. 0.3 x 5 y 8 z 0. 5 x 4 y 8 z 1.8 (0.36, 0.12, 0.06)
Solve each problem by using a system of three equations in three unknowns. See Example 5.
31. Diversification. Ann invested a total of $12,000 in stocks, bonds, and a mutual fund. She received a 10% return on her stock investment, an 8% return on her bond investment, and a 12% return on her mutual fund. Her total return was $1,230. If the total investment in stocks and bonds equaled her mutual fund investment, then how much did she invest in each? $1,500 stocks, $4,500 bonds, $6,000 mutual fund
4.3 Systems of Linear Equations in Three Variables (4-25) 223
32. Paranoia. Fearful of a bank failure, Norman split his life savings of $60,000 among three banks. He received 5%, 6%, and 7% on the three deposits. In the account earning 7% interest, he deposited twice as much as in the account earning 5% interest. If his total earnings were $3,760, then how much did he deposit in each account? $16,000 at 5%, $12,000 at 6%, $32,000 at 7% 33. Big tipper. On Monday Headley paid $1.70 for two cups of coffee and one doughnut, including the tip. On Tuesday he paid $1.65 for two doughnuts and a cup of coffee, including the tip. On Wednesday he paid $1.30 for one coffee and one doughnut, including the tip. If he always tips the same amount, then what is the amount of each item? Coffee $0.40, doughnut $0.35, tip $0. 34. Weighing in. Anna, Bob, and Chris will not disclose their weights but agree to be weighed in pairs. Anna and Bob to- gether weigh 226 pounds. Bob and Chris together weigh 210 pounds. Anna and Chris together weigh 200 pounds. How much does each student weigh? Anna 108 pounds, Bob 118 pounds, Chris 92 pounds
226
Anna & Bob Bob & Chris Anna & Chris
210 200
F I G U R E F O R E X E R C I S E 3 4
35. Lunch-box special. Salvador’s Fruit Mart sells variety packs. The small pack contains three bananas, two apples, and one orange for $1.80. The medium pack contains four bananas, three apples, and three oranges for $3.05. The family size contains six bananas, five apples, and four or- anges for $4.65. What price should Salvador charge for his lunch-box special that consists of one banana, one apple, and one orange? $0. 36. Three generations. Edwin, his father, and his grandfather have an average age of 53. One-half of his grandfather’s age, plus one-third of his father’s age, plus one-fourth of Edwin’s age is 65. If 4 years ago, Edwin’s grandfather was four times as old as Edwin, then how old are they all now? Edwin 24, father 51, grandfather 84 37. Error in the scale. Alex is using a scale that is known to have a constant error. A can of soup and a can of tuna are placed on this scale, and it reads 24 ounces. Now four iden- tical cans of soup and three identical cans of tuna are placed on an accurate scale, and a weight of 80 ounces is recorded. If two cans of tuna weigh 18 ounces on the bad scale, then what is the amount of error in the scale and what is the correct weight of each type of can? Soup 14 ounces, tuna 8 ounces, error 2 ounces
38. Three-digit number. The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the old number. If the hundreds digit plus twice the tens digit is equal to the units digit, then what is the number? 137 39. Working overtime. To make ends meet, Ms. Farnsby works three jobs. Her total income last year was $48,000. Her income from teaching was just $6,000 more than her income from house painting. Royalties from her textbook sales were one-seventh of the total money she received from teaching and house painting. How much did she make from each source last year? $24,000 teaching, $18,000 painting, $6,000 royalties 40. Pocket change. Harry has $2.25 in nickels, dimes, and quarters. If he had twice as many nickels, half as many dimes, and the same number of quarters, he would have $2.50. If he has 27 coins altogether, then how many of each does he have? 15 nickels, 10 dimes, 2 quarters
224 (4-26) Chapter 4 Systems of Linear Equations
GET TING MORE INVOLVED
41. Exploration. Draw diagrams showing the possible ways to position three planes in three-dimensional space. 42. Discussion. Make up a system of three linear equations in three variables for which the solution set is (0, 0, 0). A system with this solution set is called a homogeneous system. Why do you think it is given that name? 43. Cooperative learning. Working in groups, do parts (a)–(d) below. Then write a report on your findings. a) Find values of a , b , and c so that the graph of y ax^2 bx c goes through the points (1, 2), (1, 0), and (2, 7). b) Arbitrarily select three ordered pairs and find the equa- tion of the parabola that goes through the three points. c) Could more than one parabola pass through three given points? Give reasons for your answer. d) Explain how to pick three points for which no parabola passes through all of them.
S O L V I N G L I N E A R S Y S T E M S
U S I N G M A T R I C E S
You solved linear systems in two variables by substitution and addition in
Sections 4.1 and 4.2. Those methods are done differently on each system. In this
section you will learn the Gaussian elimination method, which is related to the
addition method. The Gaussian elimination method is performed in the same way
on every system. We first need to introduce some new terminology.
Matrices
A matrix is a rectangular array of numbers. The rows of a matrix run horizontally,
and the columns of a matrix run vertically. A matrix with m rows and n columns has
order m n (read “ m by n ”). Each number in a matrix is called an element or
entry of the matrix.
E X A M P L E 1 Order of a matrix
Determine the order of each matrix.
a) b) c) d) [1 3 6]
Solution
Because matrix (a) has 3 rows and 2 columns, its order is 3 2. Matrix (b) is a
2 2 matrix, matrix (c) is a 3 3 matrix, and matrix (d) is a 1 3 matrix.
The Augmented Matrix
The solution to a system of linear equations such as
x 2 y 5
3 x y 6
I n t h i s
s e c t i o n
(^) Matrices (^) The Augmented Matrix (^) The Gaussian Elimination Method (^) Inconsistent and Dependent Equations
s t u d y t i p
As soon as possible after class, find a quiet place and work on your homework. The longer you wait, the harder it is to remember what happened in class.
