Solve Systems of Linear Equations in Three Variables. Section 3.4.
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Solve Systems of Linear Equations in Three Variables. Section 3.4
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Transcript of Solve Systems of Linear Equations in Three Variables. Section 3.4.
Solve Systems of Linear Equations in Three Variables.
Section 3.4
A linear equation in three variables x, y, and z is an equation of the form ax + by + cz = d where a, b, and c are not all zero.
Linear Equation in Three Variables
The following is an example of a system of three linear equations.
The solution of such a system is an ordered triple (x, y, z) whose coordinates make each equation true.
System of Three Linear Equations
2 5
3 2 16
4 3 5 3
x y z
x y z
x y z
The graph of a linear equation in three variables is a plane in three-dimensional space.
The graphs of three such equations that form a system are three planes whose intersection determines the number of solutions.
Exactly one solutionThe planes intersect in a single point.
Infinitely many solutionsThe planes intersect in a line or the same plane.
No solutionThe planes have no common point of intersection.
1. Rewrite the linear system in three variables as a linear system in two variables. by using the elimination method.
2. Solve the new linear system for both of its variables.
3. Substitute the values found in #2 into one of the original equations and solve for the remaining variable.
The Elimination Method for a Three-Variable System
If you obtain a false statement such as 0 = 1, in any of the steps, then the system has not solution.
If you do not obtain a false equation, but obtain an identity such as 0 = 0, then the system has infinitely many solutions.
Solve the system.
1. Eliminate the y since it has a coefficient of -1 in the 1st equation.
Example 1
2 6 4 1
6 4 5 7 2
4 2 5 9 3
x y z
x y z
x y z
2 6 4 1
6 4 5 7 2
x y z
x y z
4 2 6 4
6 4 5 7
x y z
x y z
8 4 24 16
6 4 5 7
x y z
x y z
14 19 23x z
2 6 4 1
4 2 5 9 3
x y z
x y z
2 2 6 4
4 2 5 9
x y z
x y z
4 2 12 8
4 2 5 9
x y z
x y z
8 7 17x z
2. Solve the new two variable linear system.
14 19 23
8 7 17
x z
x z
8 14 19 23
14 8 7 17
x z
x z
112 152 184
112 98 238
x z
x z
54 54z 1z
8 7 1 17x
8 7 17x z
3x
2 6 4x y z
2 3 6 1 4y
4y
The solution is (-3, 4, 1).
3. Substitute −3 for x and 1 for z in one of the three equations and solve for y.
Solve the system.
Example 2
2 1
3 3 3 8 2
2 4 7 3
x y z
x y z
x y z
2 1
3 3 3 8 2
x y z
x y z
3 2
3 3 3 8
x y z
x y z
3 3 3 6
3 3 3 8
x y z
x y z
0 2 False
No solution
Solve the system.
Example 3
6 1
6 2
4 4 24 3
x y z
x y z
x y z
6 1
6 2
x y z
x y z
2 2 12 or 6 x z x z
6 2
4 4 24 3
x y z
x y z5 5 30 or 6 x z x z
The solution is the line x + z = 6
So there are infinitely many solutions.
Solve the system.
Example 4
3 2 10 1
6 2 2 2
4 3 7 3
x y z
x y z
x y z
3 2 10 1
6 2 2 2
x y z
x y z
2 3 2 10
6 2 2
x y z
x y z
6 2 4 20
6 2 2
x y z
x y z
12 3 18 x z
6 2 2 2
4 3 7 3
x y z
x y z
2 6 2 2
4 3 7
x y z
x y z
12 4 2 4
4 3 7
x y z
x y z
13 5 3 x z
12 3 18 x z
4 6
13 5 3
x z
x z
4 6 x z
5 4 6
13 5 3
x z
x z
20 5 30
13 5 3
x z
x z
33 33x 1x
13 5 3 x z
13 1 5 3 z
5 10z
2z
3 1 2 2 10 y
3y
(1, 3, −2)
a) Define the unknowns.
b) Set up the system of equations.
c) Solve the system of equations.
d) Write a sentence to answer the question.
System of Three Linear Equations Application Problems
A coin bank holds nickels, dimes, and quarters. There are 45 coins in the bank and the value of the coins is $4.75. If there are five more nickels than quarters, find the number of each type of coin in the bank.
Example 1
a)
N = # of nickels
D = # of dimes
Q = # of quarters
b)
We will use substitution to solve the 1st part of this problem.
N + D + Q = 45.05N + .10D + .25Q = 4.75
N = Q + 5
Q + 5 + D + Q = 45
.05 Q + 5 + .10D + .25Q = 4.75
N + D + Q = 45
.05N + .10D + .25Q = 4.75
N = Q + 5
2Q + D = 40
.30Q + .10D = 4.50
c)
2Q + D = 40
.30Q + .10D = 4.50
2Q + D = 40 .10
.30Q + .10D = 4.50
.20Q .10 D = 4
.30Q + .10D = 4.50
.10Q = .50Q = 5
2Q + D = 40
2 5 + D = 40D = 30
N + 30 + 5 = 45N = 10
d) There are 10 nickels, 30 dimes, and 5 quarters.
John invested $6500 in three different mutual funds for one year. He earned a total of $560 in simple interest on the three investments. The first fund paid 5% interest, and the second fund paid 8% interest, and the third fund paid 10% interest. If the sum of the first two investments was $500 less than the amount of the third investment, find the amount he invested at each rate.
Example 2
x = amount invested in 5% fund
y = amount invested in 8% fund
z = amount invested in 10% fund
6500 x y z
6500
5 8 10 56000
500
x y z
x y z
x y z
.05 .08 .10 560 x y z500 x y z
6500
5 8 10 56000
500
x y z
x y z
x y z
6500
500
x y z
x y z
2 2 6000x y
6500
5 8 10 56000
x y z
x y z
6500 10
5 8 10 56000
x y z
x y z
10 10 10 65000
5 8 10 56000
x y z
x y z
5 2 9000x y
2 2 6000
5 2 9000
x y
x y
3 3000x 1000x
2 1000 2 6000y 2 4000y
2000y 6500 1000 2000z 3500z
John invested $1000 at 5%, $2000 at 8%, and $3500 at 10%.
The sum of the digits of a three digit number is 12. Five times the units digit plus 6 times the tens digit is 28. If 2 times the tens digit is subtracted from 3 times the hundreds digit, the result is 15. Find the number.
Example 3
U = the units digit
T = the tens digit
H = the hundred digit
12 U T H5 6 28 U T3 2 15 H T
12
5 6 0 28
0 2 3 15
U T H
U T H
U T H
6 6 6 72
5 6 0 28
U T H
U T H
12
5 6 0 28
0 2 3 15
U T H
U T H
U T H
6 44U H 2 2 2 24
0 2 3 15
U T H
U T H
2 5 39U H
6 44
2 5 39
U H
U H
6 44 2
2 5 39
U H
U H
2 12 88
2 5 39
U H
U H
7 49H 7H
6 44U H
6 7 44U 2U
2 7 12T 3T
The number is 732.
