Solving a linear system 2 x 1 +2 x 2 +10 x 3 = 2 3 x 1 + x 2 +9 - - PowerPoint PPT Presentation

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Aug 09, 2023 178 likes •420 views

Solving a linear system 2 x 1 +2 x 2 +10 x 3 = 2 3 x 1 + x 2 +9 x 3 = 5 4 x 1 2 x 2 14 x 3 = 6 Solving a linear system 2 x 1 +2 x 2 +10 x 3 = 2 R 1 1 2 R 1 3 x 1 + x 2 +9 x 3 = 5

SLIDE 1

Solving a linear system

   −2x1 +2x2 +10x3 = 2 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

SLIDE 2

Solving a linear system

   −2x1 +2x2 +10x3 = 2 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R1← 1

2 R1

− − − − − →

SLIDE 3

Solving a linear system

   −2x1 +2x2 +10x3 = 2 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R1← 1

2 R1

− − − − − →    x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

SLIDE 4

Solving a linear system

   −2x1 +2x2 +10x3 = 2 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R1← 1

2 R1

− − − − − →    x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6   −2 2 10 2 −3 1 9 5 4 −2 −14 −6  

R1← 1

2 R1

− − − − − →   1 −1 −5 −1 −3 1 9 5 4 −2 −14 −6  

SLIDE 5

Solving a linear system

   x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

SLIDE 6

Solving a linear system

   x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R2←R2+3R1

− − − − − − − − →

SLIDE 7

Solving a linear system

   x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R2←R2+3R1

− − − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 4x1 −2x2 −14x3 = −6

SLIDE 8

Solving a linear system

   x1 −x2 −5x3 = −1 −3x1 +x2 +9x3 = 5 4x1 −2x2 −14x3 = −6

R2←R2+3R1

− − − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 4x1 −2x2 −14x3 = −6   1 −1 −5 −1 −3 1 9 5 4 −2 −14 −6  

R2←R2+3R1

− − − − − − − − →   1 −1 −5 −1 0 −2 −6 2 4 −2 −14 −6  

SLIDE 9

Solving a linear system

   x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 4x1 −2x2 −14x3 = −6

R3←R3−4R1

− − − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 2x2 +6x3 = −2

SLIDE 10

Solving a linear system

   x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 4x1 −2x2 −14x3 = −6

R3←R3−4R1

− − − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 2x2 +6x3 = −2   1 −1 −5 −1 0 −2 −6 2 4 −2 −14 −6  

R3←R3−4R1

− − − − − − − − →   1 −1 −5 −1 0 −2 −6 2 2 6 −2  

SLIDE 11

Solving a linear system

   x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 2x2 +6x3 = −2

R3←R3+R2

− − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 = 0

SLIDE 12

Solving a linear system

   x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 2x2 +6x3 = −2

R3←R3+R2

− − − − − − − →    x1 −x2 −5x3 = −1 −2x2 −6x3 = 2 = 0   1 −1 −5 −1 0 −2 −6 2 2 6 −2  

R3←R3+R2

− − − − − − − →   1 −1 −5 −1 0 −2 −6 2  

SLIDE 13

Solving a linear system

x1 −x2 −5x3 = −1 −2x2 −6x3 = 2

R3←− 1

2 R3

− − − − − − → x1 −x2 −5x3 = −1 x2 +3x3 = −1

SLIDE 14

Solving a linear system

x1 −x2 −5x3 = −1 −2x2 −6x3 = 2

R3←− 1

2 R3

− − − − − − → x1 −x2 −5x3 = −1 x2 +3x3 = −1   1 −1 −5 −1 0 −2 −6 2  

R3←− 1

2 R3

− − − − − − →   1 −1 −5 −1 1 3 −1  

SLIDE 15

Solving a linear system

x1 −x2 −5x3 = −1 x2 +3x3 = −1

R1←R1+R2

− − − − − − − → x1 −x2 −5x3 = −1 x2 +3x3 = −1

SLIDE 16

Solving a linear system

x1 −x2 −5x3 = −1 x2 +3x3 = −1

R1←R1+R2

− − − − − − − → x1 −x2 −5x3 = −1 x2 +3x3 = −1   1 −1 −5 −1 1 3 −1  

R1←R1+R2

− − − − − − − →   1 0 −2 −2 0 1 3 −1 0 0  

SLIDE 17

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

SLIDE 18

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1

SLIDE 19

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1 = ⇒ x1 = 2x3 − 2 x2 = −3x3 − 1 = ⇒   x1 x2 x3  

SLIDE 20

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1 = ⇒ x1 = 2x3 − 2 x2 = −3x3 − 1 = ⇒   x1 x2 x3   =   2t − 2 −3t − 1 t  

SLIDE 21

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1 = ⇒ x1 = 2x3 − 2 x2 = −3x3 − 1 = ⇒   x1 x2 x3   =   2t − 2 −3t − 1 t   =   2t −3t t  +   −2 −1  

SLIDE 22

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1 = ⇒ x1 = 2x3 − 2 x2 = −3x3 − 1 = ⇒   x1 x2 x3   =   2t − 2 −3t − 1 t   =   2t −3t t  +   −2 −1   =   2 −3 1   t+   −2 −1  

SLIDE 23

Solving a linear system

  1 0 −2 −2 0 1 3 −1 0 0  

−2x3 = −2 x2 +3x3 = −1 = ⇒ x1 = 2x3 − 2 x2 = −3x3 − 1 = ⇒   x1 x2 x3   =   2t − 2 −3t − 1 t   =   2t −3t t  +   −2 −1   =   2 −3 1   t+   −2 −1   (Parametric form)