1.1
SOLUTIONS
Notes: The key exercises are 7 (or 11 or 12), 19?22, and 25. For brevity, the symbols R1, R2,¡¦, stand for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section.
1.
x1 + 5 x2 = 7 ?2 x1 ? 7 x2 = ?5
? 1 ? ?2 ?
5 ?7
7? ?5? ?
x1 + 5 x2 = 7
Replace R2 by R2 + (2)R1 and obtain:
3x2 = 9
x1 + 5 x2 = 7 x2 = 3 x1
?1 ?0 ? ?1 ?0 ? ?1 ?0 ?
5 3 5 1 0 1
7? 9? ? 7? 3? ? ?8? 3? ?
Scale R2 by 1/3: Replace R1 by R1 + (?5)R2: The solution is (x1, x2) = (?8, 3), or simply (?8, 3).
2.
= ?8
x2 = 3
2 x1 + 4 x2 = ?4 5 x1 + 7 x2 = 11
?2 ?5 ?
4 7
?4 ? 11 ? ?
x1 + 2 x2 = ?2
Scale R1 by 1/2 and obt¡¦(»ý·«)
ain: Replace R2 by R2 + (?5)R1: Scale R2 by ?1/3: Replace R1 by R1 + (?2)R2: The solution is (x1, x2) = (12, ?7), or simply (12, ?7).
5 x1 + 7 x2 = 11
x1 + 2 x2 = ?2
?1 ?5 ? ?1 ?0 ? ?1 ?0 ? ?1 ?0 ?
2 7 2 ?3 2 1 0 1
?2 ? 11 ? ? ?2 ? 21? ? ?2 ? ?7 ? ? 12 ? ?7 ? ?
?3x2 = 21
x1 + 2 x2 =
