1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

2. Chapter 4 Systems of Equations

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 4.4 Solving Systems of Equations by Matrices

4. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall In this section we introduce solving a system of equations by a matrix. A matrix is a rectangular array of numbers. The numbers aligned horizontally are in the same row. The numbers aligned vertically are in the same column. The matrix has 2 rows and 3 columns It is called a 2  3 matrix.

5. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall A matrix can be created from the coefficients of a system of equations. System of EquationsCorresponding Matrix 2x + 3y = – 4 x – 5y = 8 Each number in a matrix is called an element. The following row operations can be performed on matrices, and the result is an equivalent matrix.

6. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Elementary Row Operations 1. Any two rows in a matrix may be interchanged. 2. The elements of any row may be multiplied (or divided) by the same nonzero number. 3. The elements of any row may be multiplied (or divided) by a nonzero number and added to their corresponding elements in any other row.

7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Our goal in using matrices to solve the system of equations is to get the matrix into the form Then the last row corresponds to 0·x + 1·y = c or y = c, which we can use to substitute into the first equation, 1·x + a·y = b or x = b – a·y.

8. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 1 Use matrices to solve the system. Solution The corresponding matrix is: We reverse rows 1 and 2 to get 1 in the upper left. Multiply row 1 by – 2 and add to row 2. We only change row 2. row 1 element row 2 element row 1 element row 2 element row 1 element row 2 element continued

9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to solve the system. Divide row 2 by 7. The last matrix corresponds to the system To find x, we let y = 2 in the first equation, x + 3y = 11. The ordered pair solution is (5, 2). The check is left to the student.

10. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 2 Use matrices to solve the system. Solution The corresponding matrix is: Multiply row 1 by 3 and add to row 2. The equation 0 = 27 is false for all y or x values, the system is inconsistent and has no solution.

11. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Use matrices to solve the system. Solution The corresponding matrix is: Multiply row 1 by 2 and add to row 2. The equation 0 = 0 is true for all y or x values, the lines are the same. Choose either line as the representing equation for the solution.

12. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall This same methodology can be extended to a linear system of three equations with 3 variables. Now the matrix will have 3 rows and 4 columns. We will still want zeros in the lower left corner and ones along the diagonal from upper left to lower right.

13. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example 3 Use matrices to solve the system. Solution The corresponding matrix is: Add row 1 and row 3.

14. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to solve the system. Multiply row 1 by  3 and add to row 2. Add row 3 to row 2.

15. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to solve the system. Multiply row 2 by  1. Multiply row 2 by  3 and add to row 3.

16. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to solve the system. Divide row 3 by  14. The matrix corresponds to the system Replace z with 4 in the second equation and solve for y.

17. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use matrices to solve the system. To find x, we let z = 4, y = 5 in the first equation. The ordered triple is (0, 5, 4). Check to see that it satisfies all three equations in the original system.