The document provides examples for solving systems of equations using matrices. It explains how to set up the coefficient matrix, variable matrix, and constant matrix. It then shows how to multiply the inverse of the coefficient matrix to isolate the variable matrix and solve the system. Several word problems involving investments are presented with their corresponding systems of equations and matrix solutions. Students are assigned practice problems solving systems of equations using the matrix method.

1. In other words… Solve the system of equations: Y = 2x -6 2x + 4y = -16 Find the product: Find the inverse: Write the equation of the line perpendicular to the given line y = -7x -4 and through the point (0,7).

2. 4.4 Solving System of Equations Use matrices to solve systems of linear equations in mathematical and real world situations.

3. Why? You learned to solve systems of linear equations using graphing, substitution and elimination. While there are advantages and disadvantages with these methods, you can also solve systems using the matrix method.

4. Solving Systems by matrices. A financial manager wants to invest 50,000 for a client by putting some of the money in a low risk investment that earns 5% per year and some of the money in high risk investment that earns 14% per year. How much money should be invested in each to earn 5000 in interest per year? Let x represent amount invested at 5%. Let y represent amount invested at 14%. Write a system of equations.

5. Investments x + y = 50,000 .05x + .14y = 5000 To write a matrix equation Make a coefficient matrix: Make a variable matrix: Make a constant matrix: A X = B

6. Undo… To undo multiplication,use the inverse matrix of A. A (A -1 ) X=(B) (A -1 ) A matrix A times its inverse (A -1 ) produces an identity . When X is multiplied by the Identity matrix it is the same as multiplying by 1. I X=(B) (A -1 )

7. Solving the Matrix Enter the coefficient matrix and the constant matrix into your calculator. The manager should invest 22,222.22 at 5% and 27,777.78 at 14% to achieve the earned income goal of $5000 interest per year.

8. Investing How much money should the banker invest at each interest rate to earn $4000 per year? Josh has earned $5000.00 in the past 3 summers. Now he want to invest the money at 4% per year and 11% per year. How much should he invest at each interest rate to earn 500 in interest per year? Write a system of equations and solve it using a matrix equation.

9. Solving a system in 3 variables. Refer to the system at the right. Notice there are 3 variables x,y, and z. Write the system as a matrix equation and solve. Begin by writing each equation in standard form:

10. Write the system as a matrix equation. Coefficient Matrix: Variable Matrix: Constant Matrix:

11. Multiply by the Inverse A(X) = B A(X) (A -1 ) = B (A -1 ) X = B (A -1 )

12. Solution Check the solution in the original system by substituting in for x , y, and z. If x = -2, y = 4 and z = 5 Then…

13. Solve the System Use the Matrix Method to solve:

14. Check If x = 1, y = -1 and z = -3 Then: The solution to the system is 1, -1 , -3.

15. Try this… Write the system as a matrix equation Solve the matrix.

16. Dependant or Inconsistent Systems Not all systems have unique solutions. In a matrix equation in the form A X = B If the coefficient matrix A does not have an inverse, The system does not have a unique solution. Solve by writing a matrix equation: Dependant: infinity many solutions, line on top of line. Inconsistent: no solution, lines parallel

17. questions? Assignment 4.4 # 12,15, 18, 19, 25,26, 28, 30, 33, 34, 36. grrrrrrrrrrrr