Mat 150 – Class #21

Objectives

Solve systems of linear equations in three variables using left-to-right elimination

Find solutions of dependent systemsDetermine when a system of equations is inconsistent

Operations on a System of EquationsThe following operations result in an equivalent system.1. Interchange any two equations.2. Multiply both sides of any equation by the same

nonzero number.3. Multiply any equation by a number, add the result to a

second equation, and then replace the second equation with the sum.

Left-to-Right Elimination Method of Solving Systemsof Linear Equations in Three Variables x, y, and z1. If necessary, interchange two equations or use

multiplication to make the coefficient of x in the first equation a 1.

2. Add a multiple of the first equation to each of the following equations so that the coefficients of x in the second and third equations become 0.

3. Multiply (or divide) both sides of the second equation by a number that makes the coefficient of y in the second equation equal to 1.

4. Add a multiple of the (new) second equation to the (new) third equation so that the coefficient of y in the newest third equation becomes 0.

Left-to-Right Elimination Method of Solving Systemsof Linear Equations in Three Variables x, y, and z

5. Multiply (or divide) both sides of the third equation by a number that makes the coefficient of z in the third equation equal to 1. This gives the solution for z in the system of equations.

6. Use the solution for z to solve for y in the second equation. Then substitute values for y and z to solve for x in the first equation. (This is called back substitution.)

Example

Solve the system

Solution

2 3 1

2 3

3 9

x y z

x y z

x y z

Example (cont)

Example

A manufacturer of furniture has three models of chairs: Anderson, Blake, and Colonial. The numbers of hours required for framing, upholstery, and finishing for each type of chair are given in the table. The company has 1500 hours per week for framing, 2100 hours for upholstery, and 850 hours for finishing. How many of each type of chair can be produced under these conditions?

Example (cont)

Let x represent AndersonLet y represent BlakeLet z represent Colonial

Example (cont)

Nonunique Solutions

It is not always possible to reduce a system of three equations in three variables to a system in which the third equation contains one variable.

A system in which 0 = x (any number) is __________________.

A system in which 0 = 0 is a ______________________.

Example

Solve the system.

4 5

3 0

4 20

x y z

x z

x y z

Example

Solve the system.

2 0

2 6

2 6

x y z

x y z

x y z

Example (cont)

Assignment

Pg. 505-508#1-25 odd#27, 31, 34,