Section 3.6 – Solving Systems Using Matrices Students will be able to: Represent a system of linear equations with a matrix To solve a system of linear equations using matrices NEW VOCABULARY: Matrix Matrix Element Row Operations

Section 3.6 – Solving Systems Using Matrices

Section 3.6 – Solving Systems Using Matrices An array of numbers, such as each of those suggested by the tile arrangements in the Solve It, is a matrix. Essential Understanding You can use a matrix to represent and solve a system of equations without writing ANY variables. REAL LIFE

Section 3.6 – Solving Systems Using Matrices A matrix is a rectangular array of numbers. You usually display the array with brackets. The dimensions of a matrix are the number of rows and columns in the array. Each number is a matrix is a matrix element. You can identify a matrix element by its row and column of numbers. In Matrix A, a12 is the element in row 1 and column 2. a12 is the element 4.

Section 3.6 – Solving Systems Using Matrices You can represent a system of equations efficiently with a matrix. Each matrix row represents an equation. The last matrix column shows the constants to the right of the equal signs. Each of the other columns shows the coefficients of one of the variables.

Section 3.6 – Solving Systems Using Matrices

Section 3.6 – Solving Systems Using Matrices Problem 1: How can you represent the system of equations with a matrix? 2x + y = 9 x – 6y = -1

Section 3.6 – Solving Systems Using Matrices Problem 1: How can you represent the system of equations with a matrix? x – 3y + z = 6 x + 3z = 12 y = -5x + 1

Section 3.6 – Solving Systems Using Matrices Problem 1: How can you represent the system of equations with a matrix? -4x – 2y = 7 3x + y = -5

Section 3.6 – Solving Systems Using Matrices Problem 1: How can you represent the system of equations with a matrix? 4x – y + 2z = 1 y + 5z = 20 2x = -y + 7

Section 3.6 – Solving Systems Using Matrices Problem 2: What linear system of equations does this matrix represent?

Section 3.6 – Solving Systems Using Matrices Problem 2: What linear system of equations does this matrix represent?

Section 3.6 – Solving Systems Using Matrices You can use a matrix that represents a system of equations to solve the system. In this way, you do not have to write the variables. To solve the system using a matrix, use the steps for solving by elimination. Each step is a row operation.

Section 3.6 – Solving Systems Using Matrices Your goal is to use row operations to get a matrix in the form: Notice that the first matrix represents the system x = a, y = b, which then will be the solution of a system of two equations in two unknowns. The second matrix represents the system x = a, y = b, and z = c.

Section 3.6 – Solving Systems Using Matrices

Section 3.6 – Solving Systems Using Matrices Problem 3: What is the solution of the system? x + 4y = -1 2x + 5y = 4

Section 3.6 – Solving Systems Using Matrices Problem 3: What is the solution of the system? x + y = 5 -2x + 4y = 8

Section 3.6 – Solving Systems Using Matrices Problem 3: What is the solution of the system? x +3y = 22 2x - y = 2

Section 3.6 – Solving Systems Using Matrices Matrices that represent the solution of a system are in reduced row echelon form. Many calculators have a rref(reduced row echelon form) function for working with matrices. This function will do all the row operations for you. You can use the rref to solve a system of equations.

Section 3.6 – Solving Systems Using Matrices What is the solution of the system of equations? 2a + 3b – c = 1 -4a + 9b + 2c = 8 -2a + 2c = 3

Section 3.6 – Solving Systems Using Matrices What is the solution of the system of equations? a + 4b + 6c = 21 2a – 2b + c = 4 -8b + c = -1

Section 3.6 – Solving Systems Using Matrices