Issuu

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 𝑎12 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏2 ⋮

⋮

𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 𝑏𝑚 . 𝑘≠0

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 (𝑘𝑎12 )𝑥1 + (𝑘𝑎22 )𝑥2 + ⋯ + (𝑘𝑎2𝑛 )𝑥𝑛 = 𝑘𝑏2 ⋮

⋮

𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 𝑏𝑚

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 (𝑘𝑎12 )𝑥1 + (𝑘𝑎22 )𝑥2 + ⋯ + (𝑘𝑎2𝑛 )𝑥𝑛 = 𝑘𝑏2 ⋮

⋮

(𝑘𝑎12 + 𝑎𝑚1 )𝑥1 + (𝑘𝑎22 + 𝑎𝑚2 )𝑥2 + ⋯ + (𝑘𝑎2𝑛 + 𝑎𝑚𝑛 )𝑥𝑛 = 𝑏𝑚

1 𝑘

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 𝑎12 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏2 ⋮

⋮

(𝑘𝑎12 + 𝑎𝑚1 )𝑥1 + (𝑘𝑎22 + 𝑎𝑚2 )𝑥2 + ⋯ + (𝑘𝑎2𝑛 + 𝑎𝑚𝑛 )𝑥𝑛 = 𝑏𝑚

𝑘 𝑘

𝑎11 𝑎12 … 𝑎1𝑛 𝑎 𝑎 … 𝑎2𝑛 [ 21 22 ⋮ 𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛

𝑏1 𝑏2 ] ⋮ 𝑏𝑚

𝑎11 ≠ 0 𝑥1

𝑎𝑖1 ≠ 0

𝑎11 1 𝑎12 ′ … 𝑎1𝑛 ′ 𝑎 𝑎 … 𝑎2𝑛 [ 21 22 ⋮ 𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛

𝑏1 ′ 𝑏2 ] ⋮ 𝑏𝑚 𝑥1

𝑚−1 𝑎´21 , 𝑎´31 , … , 𝑎´𝑚1 −𝑎21 −𝑎31

𝑚 −𝑎𝑚1

1 𝑎12 ′ … 𝑎1𝑛 ′ 0 𝑎22 ′ … 𝑎2𝑛 ′ [ ⋮ 0 𝑎𝑚2 ′ … 𝑎𝑚𝑛 ′

𝑏1 ′ 𝑏2 ′ ] ⋮ 𝑏𝑚 ′

𝑚−1

𝑎22 ′ 𝑎32 ′

𝑎𝑚2 ′

𝑥1 + 3𝑥2 − 𝑥3 + 2𝑥4 = 1 5𝑥1 + 15𝑥2 + 4𝑥3 − 𝑥4 = −2 3𝑥1 + 9𝑥2 + 𝑥3 + 7𝑥4 = 5

𝑥1 + 3𝑥2 − 𝑥3 + 2𝑥4 = 1 9𝑥3 − 11𝑥4 = −7 4𝑥3 + 𝑥4 = 2 𝑥2

𝑥𝑛 𝑥𝑛−1

2𝑤 + 2𝑥 + 4𝑦 − 6𝑧 = −6 2𝑤 + 3𝑥 + 9𝑦 − 4𝑧 = 18 4𝑤 + 8𝑥 + 32𝑦 − 8𝑧 = 80 3𝑤 + 4𝑥 + 11𝑦 − 6𝑧 = 19

𝑏𝑛

𝑤 + 𝑥 + 2𝑦 − 3𝑧 = −3 𝑥 + 5𝑦 + 2𝑧 = 24 𝑦 − 𝑧 = −1 𝑧=4

𝑧=4

𝑦=3 𝑤 = 2, 𝑥 = 1, 𝑦 = 3, 𝑧 = 4.

𝑤 − 𝑥 + 3𝑦 − 𝑧 = −6 4𝑤 − 3𝑥 + 13𝑦 − 2𝑧 = −22 𝑤 + 𝑥 + 6𝑦 + 𝑧 = −3 −2𝑤 + 3𝑥 − 4𝑦 + 2𝑧 = 13

𝑤

𝑤 − 𝑥 + 3𝑦 − 𝑧 = −6 𝑥 + 𝑦 + 2𝑧 = 2 𝑦 − 2𝑧 = −1 0𝑤 + 0𝑥 + 0𝑦 + 0𝑧 = 0

𝑧 𝑤 = −9𝑧,

𝑥 = 3 − 4𝑧,

𝑦 = 2𝑧 − 1, 𝑧

𝑧=𝑧

2𝑥 + 4𝑦 + 6𝑧 = 1 3𝑥 + 7𝑦 + 2𝑧 = 5 5𝑥 + 11𝑦 + 8𝑧 = 2

1 2 7 𝑦 − 7𝑧 = 2 0𝑥 + 0𝑦 + 0𝑧 = −4 𝑥 + 2𝑦 + 3𝑧 =

𝑥, 𝑦, 𝑧

0𝑥 + 0𝑦 + 0𝑧 = −4

1 2 7 𝑦 − 7𝑧 = 2

𝑥 + 2𝑦 + 3𝑧 =

𝑦 − 7𝑧 = −

𝑦, 𝑧

1 2

𝑥1 + 𝛼12 𝑥2 + ⋯ + 𝛼1𝑛 𝑥𝑛 = 𝛽1 ⋮ 𝑥𝑖 + 𝛾𝑖−1 𝑥𝑖−1 + ⋯ + 𝛾𝑛 𝑥𝑛 = 𝛽𝑘 𝑥𝑖 + 𝛾𝑖−1 𝑥𝑖−1 + ⋯ + 𝛾𝑛 𝑥𝑛 = 𝛽𝑘+1