1 Section 5.3 Linear Systems of Equations

2 THREE EQUATIONS WITH THREE VARIABLES Consider the linear system of three equations below with three unknowns (variables). A solution of this system is simply a triple x, y, z of numbers that satisfy all three equations.

3 SOLVING A THREE EQUATION SYSTEM The method of elimination. The method of determinants. The method of matrices. We will discuss three methods for solving a three equation system. (These methods are three of the ones discussed in the last section.) NOTE: Graphs are not useful here because the graph of a linear equation with three variables is a plane. Thus, it requires graphing in 3 dimensions.

4 METHOD OF ELIMINATION 1.Eliminate x from two of the pairs of equations. (This leaves two equations with variables y and z.) 2.Use the elimination method for two variables (discussed in Section 5.2) to solve the two resulting equations in Step 1. 3.Substitute the values of y and z into one of the original equations to find the value for x. Note that there are three pairs of equations: (1) the first and second, (2) the first and third, and (3) the second and third. To use eliminations we

5 3 × 3 DETERMINANT To compute the value of the 3 × 3 coefficient determinant:

6 DETERMINANT METHOD The determinant solution to the system of equations is where Δ is the coefficient determinant.

7 MATRIX METHOD For the system let Then the solution is