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# Ch4.4 Systems W Matrices

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solving systems with matrices

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### Ch4.4 Systems W Matrices

1. 1. In other words… <ul><li>Solve the system of equations: </li></ul><ul><ul><li>Y = 2x -6 </li></ul></ul><ul><ul><li>2x + 4y = -16 </li></ul></ul><ul><li>Find the product: </li></ul><ul><li>Find the inverse: </li></ul><ul><li>Write the equation of the line perpendicular to the given line </li></ul><ul><ul><li>y = -7x -4 and through the point (0,7). </li></ul></ul>
2. 2. 4.4 Solving System of Equations Use matrices to solve systems of linear equations in mathematical and real world situations.
3. 3. Why? <ul><li>You learned to solve systems of linear equations using graphing, substitution and elimination. </li></ul><ul><li>While there are advantages and disadvantages with these methods, you can also solve systems using the matrix method. </li></ul>
4. 4. Solving Systems by matrices. <ul><li>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. </li></ul><ul><li>How much money should be invested in each to earn 5000 in interest per year? </li></ul><ul><li>Let x represent amount invested at 5%. </li></ul><ul><li>Let y represent amount invested at 14%. </li></ul><ul><li>Write a system of equations. </li></ul>
5. 5. Investments <ul><li>x + y = 50,000 </li></ul><ul><li>.05x + .14y = 5000 </li></ul><ul><li>To write a matrix equation </li></ul><ul><ul><li>Make a coefficient matrix: </li></ul></ul><ul><ul><li>Make a variable matrix: </li></ul></ul><ul><ul><li>Make a constant matrix: </li></ul></ul><ul><ul><li>A X = B </li></ul></ul>
6. 6. Undo… <ul><li>To undo multiplication,use the inverse matrix of A. </li></ul><ul><li>A (A -1 ) X=(B) (A -1 ) </li></ul><ul><li>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. </li></ul><ul><li>I X=(B) (A -1 ) </li></ul>
7. 7. Solving the Matrix <ul><li>Enter the coefficient matrix and the constant matrix into your calculator. </li></ul><ul><li>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. </li></ul>
8. 8. Investing <ul><li>How much money should the banker invest at each interest rate to earn \$4000 per year? </li></ul><ul><li>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? </li></ul><ul><li>Write a system of equations and solve it using a matrix equation. </li></ul>
9. 9. Solving a system in 3 variables. <ul><li>Refer to the system at the right. </li></ul><ul><li>Notice there are 3 variables </li></ul><ul><ul><li>x,y, and z. </li></ul></ul><ul><li>Write the system as a matrix equation and solve. </li></ul><ul><li>Begin by writing each equation in standard form: </li></ul>
10. 10. Write the system as a matrix equation. <ul><li>Coefficient Matrix: </li></ul><ul><li>Variable Matrix: </li></ul><ul><li>Constant Matrix: </li></ul>
11. 11. Multiply by the Inverse <ul><li>A(X) = B </li></ul><ul><li>A(X) (A -1 ) = B (A -1 ) </li></ul><ul><li>X = B (A -1 ) </li></ul>
12. 12. Solution <ul><li>Check the solution in the original system by substituting in for x , y, and z. </li></ul><ul><li>If x = -2, y = 4 and z = 5 </li></ul><ul><li>Then… </li></ul>
13. 13. Solve the System <ul><li>Use the Matrix Method to solve: </li></ul>
14. 14. Check <ul><li>If x = 1, y = -1 and z = -3 </li></ul><ul><li>Then: </li></ul><ul><li>The solution to the system is 1, -1 , -3. </li></ul>
15. 15. Try this… <ul><li>Write the system as a matrix equation </li></ul><ul><li>Solve the matrix. </li></ul>
16. 16. Dependant or Inconsistent Systems <ul><li>Not all systems have unique solutions. </li></ul><ul><li>In a matrix equation in the form A X = B </li></ul><ul><ul><li>If the coefficient matrix A does not have an inverse, </li></ul></ul><ul><ul><li>The system does not have a unique solution. </li></ul></ul><ul><ul><li>Solve by writing a matrix equation: </li></ul></ul><ul><ul><li>Dependant: infinity many solutions, line on top of line. </li></ul></ul><ul><ul><li>Inconsistent: no solution, lines parallel </li></ul></ul>
17. 17. questions? Assignment 4.4 # 12,15, 18, 19, 25,26, 28, 30, 33, 34, 36. grrrrrrrrrrrr