6 minutes Warm-Up Find each product.

4.4 Solving Systems with Matrix Equations Objectives: Use matrices to solve systems of linear equations in mathematical and real-world situations

Example 1 Amy has saved $5000 during the past three summers. Now she wants to put some of the money into an investment that earns 4% per year and some into an investment that earns 11% per year. How much money should Amy invest at each interest rate to earn $500 in interest per year? Let x represent the amount invested at 4% Step 3: Write the system as a matrix equation, AX = B Step 4: Solve for the variable matrix, X, by finding the product A-1B Step 2: Write a system of equations Step 1: Choose your variables x + y = 5000 .04x + .11y = 500 Let y represent the amount invested at 11% $714.29 at 4% and $4285.71 at 11%

Example 2 Refer to the system of equations below. a) Write the system as a matrix equation b) Solve the matrix equation x = - 1.8 y = - 0.1 z = - 0.8

Example 3 Refer to the system of equations below. a) Write the system as a matrix equation b) Solve the matrix equation

Practice Refer to the system of equations below. a) Write the system as a matrix equation b) Solve the matrix equation

6 minutes Warm-Up Write each system as a matrix equation. Then solve the system, if possible, by using the matrix equation.

4.5.1 Using Matrix Row Operations Objectives: Represent a system of equations as an augmented matrix Perform elementary row operations on matrices

Matrix Row Operations The row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent. The row-reduction method of solving a system is performed on an augmented matrix. An augmented matrix consists of the coefficients and constant terms in the system of equations. System Augmented Matrix

Matrix Row Operations The goal of the row-reduction method is to transform, if possible, the coefficient columns into columns that form an identity matrix. This is called the reduced row-echelon form of an augmented matrix if the matrix represents an independent system. The resulting constants will represent the unique solution to the system.

Example 1 Solve the system of equations by using the row-reduction method. Then classify the system. 1 2 16 -3 -21 x = 2; y = 7 independent

Practice Solve the system of equations by using the row-reduction method. Then classify the system.

Example 2 Solve the system of equations by using the row-reduction method. Then classify the system. x – 1.4z = 0 y – 0.2z = 0 0 = 1 no solution, inconsistent

Example 3 Solve the system of equations by using the row-reduction method. Then classify the system. x – z = -1 y + 2z = 3 0 = 0 infinitely many solutions dependent

Practice Solve the system of equations by using the row-reduction method. Then classify the system.