6.3. MULTIVARIABLE LINEAR SYSTEMS

Transcription

1 6.3. MULTIVARIABLE LINEAR SYSTEMS

2 What You Should Learn Use back-substitution to solve linear systems in row-echelon form. Use Gaussian elimination to solve systems of linear equations. Solve nonsquare systems of linear equations. Use systems of linear equations in three or more variables to model and solve real-life problems.

3 Row-Echelon Form and Back-Substitution The method of elimination can be applied to a system of linear equations in more than two variables. In fact, this method easily adapts to computer use for solving linear systems with dozens of variables. When elimination is used to solve a system of linear equations, the goal is to rewrite the system in a form to which back-substitution can be applied.

4 Row-Echelon Form and Back-Substitution Consider the following two systems of linear equations. System of Three Linear Equations in Three Variables: x 2y + 3z = 9 x + 3y = 4 2x 5y + 5z = 17 Equivalent System in Row-Echelon Form: x 2y + 3z = 9 y + 3z = 5 z = 2

5 Row-Echelon Form and Back-Substitution The second system is said to be in row-echelon form, which means that it has a stair-step pattern with leading coefficients of 1. After comparing the two systems, it should be clear that it is easier to solve the system in row-echelon form, using back-substitution.

6 Example Using Back-Substitution in Row-Echelon Form Solve the system of linear equations. x 2y + 3z = 9 y + 3z = 5 z = 2 Equation 1 Equation 2 Equation 3

7 Example Solution From Equation 3, you know the value of z. To solve for y, substitute z = 2 into Equation 2 to obtain y + 3(2) = 5 y = 1. Substitute 2 for z. Solve for y.

8 Example Solution cont d Then substitute y = 1 and z = 2 into Equation 1 to obtain x 2( 1) + 3(2) = 9 x = 1. Substitute 1 for y and 2 for z. Solve for x. The solution is x = 1, y = 1, and z = 2, which can be written as the ordered triple (1, 1, 2).

9 Gaussian Elimination Two systems of equations are equivalent if they have the same solution set. To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in row-echelon form by using the following operations.

10 Gaussian Elimination Rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of which is obtained by using one of the three basic row operations. This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss.

11 Example Using Gaussian Elimination to Solve a System Solve the system of linear equations. x 2y + 3z = 9 x + 3y = 4 2x 5y + 5z = 17 Equation 1 Equation 2 Equation 3

12 Example Solution Because the leading coefficient of the first equation is 1, you can begin by saving the x at the upper left and eliminating the other x-terms from the first column. Write Equation 1. Write Equation 2. Add Equation 1 to Equation 2.

13 Example Solution cont d x 2y + 3z = 9 y + 3z = 5 2x 5y + 5z = 17 Multiply Equation 1 by 2. Write Equation 3. Add revised Equation 1 to Equation 3.

14 Example Solution cont d x 2y + 3z = 9 y + 3z = 5 y z = 1 Now that all but the first x have been eliminated from the first column, go to work on the second column. (You need to eliminate y from the third equation.) x 2y + 3z = 9 y + 3z = 5 2z = 4

15 Example Solution cont d Finally, you need a coefficient of 1 for z in the third equation. x 2y + 3z = 9 y + 3z = 5 z = 2 This is the same system that was solved in Example 1, and, as in that example, you can conclude that the solution is x = 1, y = 1, and z = 2.

16 Gaussian Elimination As with a system of linear equations in two variables, the solution(s) of a system of linear equations in more than two variables must fall into one of three categories. We have learned that a system of two linear equations in two variables can be represented graphically as a pair of lines that are intersecting, coincident, or parallel.

17 Gaussian Elimination A system of three linear equations in three variables has a similar graphical representation it can be represented as three planes in space that intersect in one point (exactly one solution) [see Figure 6.12], intersect in a line or a plane (infinitely many solutions) [see Figures 6.13 and 6.14], or have no points common to all three planes (no solution) [see Figures 6.15 and 6.16]. Solution: one point Solution: one line Solution: one plane Solution: none Solution: none Figure 6.12 Figure 6.13 Figure 6.14 Figure 6.15 Figure 6.16