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

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 solution The planes intersect in a single point. Infinitely many solutions The planes intersect in a line or the same plane.

No solution The 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 1 st equation. Example 1

2.Solve the new two variable linear system.

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

False No solution

Solve the system. Example 3

The solution is the line x + z = 6 So there are infinitely many solutions.

Solve the system. Example 4

(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 1 st part of this problem.

c)

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

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

The number is 732.