Multivariable Linear Systems Chapter 7 Sec 3a Multivariable Linear Systems

How do you solve systems of equations with more than two variables? Essential Question How do you solve systems of equations with more than two variables? Key Vocabulary: Dependent/ Independent

Row-Echelon Form and Back Substitution System of Three Linear Equations in Three Variables Equivalent System in Row-Echelon Form This 2nd system is row-echelon, which means it has a stair step pattern with leading coefficients of 1.

Example 1: Use Back-substitution in Row-Echelon form Solve the system of linear equations. From Equationv3, you know z. To solve for y, substitute z = 2 in Equation 2. y + 4(2) = 7 … y = –1 Finally substitute y = –1 and z = 2 into Equation 1, x – 2(–1) + 3(2) = 9 … x = 1 We now can write our solution as an ordered triple (1, –1, 2) Equation 1 Equation 2 Equation 3

Two system of equations are equivalent if they have the same solution. Gaussian Elimination Two system of equations are equivalent if they have the same solution. To solve a system not in row-echelon form, first convert it to a equivalent system that is in row-echelon form by using one or more of the elementary row operations. This process is called Gaussian elimination, after Carl Friedrich Gauss (1777 – 1855). Elementary Row Operations for Systems of Equations Interchange two equations. Multiply one of the equations by a non-zero constant. Add a multiple of one equation to another

Example 2: Use Gaussian Elimination Solve the system of linear equations. Because the leading coefficient of the first equation is 1, begin by eliminating the other x terms from the first column. Now use back substitution to solve for (1, –1, 2). Equation 1 Equation 2 Equation 3 Adding the Equations 1 & 2 give new Equation 2 Adding Eq. 2 & 3 give new Equation 3 Multiply Equation 3 by 1/3 gives… Adding –2 x Eq. 1 to Eq 3 give new Equation 3

Example 3: Inconsistent System Solve the system of linear equations. Solution. Equation 1 Equation 2 Equation 3 Adding –1 x Eq. 2 to Eq 3 give new Equation 3 Adding –2 x Eq. 1 to Eq 2 give new Equation 2 Because 0 = –2 is a false statement, you can conclude that this system is inconsistent and has no solution. Adding –1 x Eq. 1 to Eq 3 give new Equation 3

A consistent system with exactly one solution is independent. Numbers of Solutions Number of solutions of a Linear System For a system of linear equations, exactly one of the following is true. There is exactly one solution.. There are infinite many solutions. (true statement) There is no solution. (false statement as in previous example) A system of linear equations is called consistent if it has at least one solution. A consistent system with exactly one solution is independent. A consistent system with infinite many solutions is dependent. A system of linear equations is called inconsistent if it has no solution.

Example 4: Infinite Solutions Solve the system of linear equations. Solution. Equation 1 Equation 2 Equation 3 Adding Eq. 1 to Eq 3 give new Equation 3 Because 0 = 0 is a true statement, you have infinite many solutions.

Example 4: Infinite Solutions We now have the equivalent system. Solve last equation in terms of z to obtain y = z. Back substituting for y produces x = 2z – 1. Finally let z = a, where a is a real number, we get. x = 2a – 1, y = a, and z = a. So, every ordered triple of the form (2a – 1, a, a) is a solution of the system.

Systems of Linear Equations in Three Variables

How do you solve systems of equations with more than two variables? Essential Question How do you solve systems of equations with more than two variables?

Chapter 7 Section 3a Text Book Daily Assignment Pg 505 – 506 #1 – 29 Odd Show all work for credit.