Section 6-1: Multivariate Linear Systems and Row Operations A multivariate linear system (also multivariable linear system) is a system of linear equations with two or more variables. ◦ Solved by:  Graphing  Substitution  Elimination

Section 6-1: Multivariate Linear Systems and Row Operations Substitution and Elimination methods can be used to convert a multivariate system into a triangular or row- echelon form.

Section 6-1: Multivariate Linear Systems and Row Operations Gaussian elimination is the algorithm used to transform a system of linear equations into an equivalent system in row-echelon form.

Section 6-1: Multivariate Linear Systems and Row Operations Example: Write the system of equations in triangular form using Gaussian elimination, then solve the system.

Section 6-1: Multivariate Linear Systems and Row Operations An augmented matrix of a system is derived from the coefficients and the constant terms of the linear equations, each written in standard form with the constant terms to the right of the equals sign. ◦ If the constant terms are not included, it is called a coefficient matrix of the system.

Section 6-1: Multivariate Linear Systems and Row Operations Example: Write an augmented matrix for the system of linear equations.

Section 6-1: Multivariate Linear Systems and Row Operations The operations used to produce equivalent systems have corresponding matrix operations. ◦ You can produce an equivalent augmented matrix. ◦ These are called elementary row operations.

Section 6-1: Multivariate Linear Systems and Row Operations

The row-echelon form of a matrix is not unique because there are many combinations of row operations that may be performed. The final solution will ALWAYS be the same.

Section 6-1: Multivariate Linear Systems and Row Operations Example: Determine whether each matrix is in row-echelon form.

Section 6-1: Multivariate Linear Systems and Row Operations If you continue to apply elementary row operations to the row-echelon form of the augmented matrix, you can obtain a matrix that resembles the following:

Section 6-1: Multivariate Linear Systems and Row Operations Solving a system to achieve this is called Gauss-Jordan elimination.

Section 6-1: Multivariate Linear Systems and Row Operations Example: Solve the system of linear equations.

Section 6-1: Multivariate Linear Systems and Row Operations When solving a system of equations, if a matrix cannot be written in reduced row-echelon form, then the system either has no solution or infinitely many solutions.

Section 6-1: Multivariate Linear Systems and Row Operations Example:

Section 6-1: Multivariate Linear Systems and Row Operations Example:

Section 6-1: Multivariate Linear Systems and Row Operations When a system has fewer equations than variables, if a matrix cannot be written in reduced row-echelon form, then the system either has no solution or infinitely many solutions.

Section 6-1: Multivariate Linear Systems and Row Operations Example:

Section 6-1: Multivariate Linear Systems and Row Operations Example:

Section 6-1: Multivariate Linear Systems and Row Operations Homework: ◦ Page 372, #3, 6, 9, 12, 18, 21, 24, 27, 32, 33 ◦ To turn in: #32