894 CHAPTER 9 Systems and Matrices 3x - 4y = 1 3132 - 4122≟1 9 - 8≟1 1 = 1 ✓ True 5x + 2y = 19 5132 + 2122≟19 15 + 4≟19 19 = 19 ✓ True True statements result, so the solution set is 513, 226. S Now Try Exercise 23. A linear system with three equations is solved in a similar way. Row transformations are used to introduce 1s down the diagonal from left to right and 0s above and below each 1. NOTE Using row operations to write a matrix in diagonal form requires effective use of the inverse properties of addition and multiplication. It is best to work in columns, beginning in each column with the element that is to become 1. In the augmented matrix c 3 5 -4 2 2 1 19d , 3 is in the first row, first column position. Use transformation 2, multiplying each entry in the first row by 1 3 (abbreviated 1 3 R12 to obtain 1 in this position. J 1 5 -4 3 2 2 1 3 19R 1 3 R1 Introduce 0 in the second row, first column by multiplying each element of the first row by -5 and adding the result to the corresponding element in the second row, using transformation 3.c 1 0 -4 3 26 3 3 1 3 52 3 d -5R1 + R2 Obtain 1 in the second row, second column by multiplying each element of the second row by 3 26, using transformation 2. J 1 0 -4 3 1 2 1 3 2R 3 26 R2 Finally, obtain 0 in the first row, second column by multiplying each element of the second row by 4 3 and adding the result to the corresponding element in the first row. c 1 0 0 1 2 3 2d 4 3 R2 + R1 This last matrix corresponds to the system x = 3 y = 2, which indicates the solution 13, 22. We can read this solution directly from the third column of the final matrix. CHECK Substitute the solution in both equations of the original system. 0 1 1 0
RkJQdWJsaXNoZXIy NjM5ODQ=