1. Solving System of Linear Equations

2. 1. Diagonal Form of a System of Equations 2. Elementary Row Operations 3. Elementary Row Operation 1 4. Elementary Row Operation 2 5. Elementary Row Operation 3 6. Gaussian Elimination Method 7. Matrix Form of an Equation 8. Using Spreadsheet to Solve System 2

3.  A system of equations is in diagonal form if each variable only appears in one equation and only one variable appears in an equation.  For example: 3

4.  Elementary row operations are operations on the equations (rows) of a system that alters the system but does not change the solutions.  Elementary row operations are often used to transform a system of equations into a diagonal system whose solution is simple to determine. 4

5.  Elementary Row Operation 1 Rearrange the equations in any order. 5

6.  Rearrange the equations of the system  so that all the equations containing x are on top. 6

7.  Elementary Row Operation 2 Multiply an equation by a nonzero number. 7

8.  Multiply the first row of the system  so that the coefficient of x is 1. 8

9.  Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation. 9

10.  Add a multiple of one row to another to change  so that only the first equation has an x term. 10

11.  Gaussian Elimination Method transforms a system of linear equations into diagonal form by repeated applications of the three elementary row operations. 1. Rearrange the equations in any order. 2. Multiply an equation by a nonzero number. 3. Change an equation by adding to it a multiple of another equation. 11

12.  Continue Gaussian Elimination to transform into diagonal form 12

13. 13

14. 14 The solution is ( x,y,z ) = (4/5,-9/5,9/5).

15.  It is often easier to do row operations if the coefficients and constants are set up in a table (matrix).  Each row represents an equation.  Each column represents a variable’s coefficients except the last which represents the constants.  Such a table is called the augmented matrix of the system of equations. 15

16.  Write the augmented matrix for the system 16 Note: The vertical line separates numbers that are on opposite sides of the equal sign.

17.  The three elementary row operations for a system of linear equations (or a matrix) are as follows:  Rearrange the equations (rows) in any order;  Multiply an equation (row) by a nonzero number;  Change an equation (row) by adding to it a multiple of another equation (row). 17