1327.sagews

EXAMPLE 1  System with No Solutions

$x -  y - 2z =  2$
$2x - 3y + 6z =  5$
$3x - 4y + 4z = 12$

M = matrix([[1, -1, -2, 2], [2, -3, 6, 5], [3, -4, 4, 12]])
M

M[1] = 2*M[0] - M[1]
M[2] = 3*M[0] - M[2]
M

M[2] = -M[1] + M[2]
M

M.echelon_form()

EXAMPLE 2  System with Infinite Solutions

$3x - 4y + 4z = 7$
$x -  y - 2z = 2$
$2x - 3y + 6z = 5$

M = matrix([[3, -4, 4, 7], [1, -1, -2, 2], [2, -3, 6, 5]])
M

M.swap_rows(0, 1)
M

M[1] = 3*M[0] - M[1]
M[2] = 2*M[0] - M[2]
M

M[2] = -M[1] + M[2]
M

var('x y z')

y = -1 + 10*z
x = 1 + 12*z

(x, y, z)

M = matrix([[3, -4, 4, 7], [1, -1, -2, 2], [2, -3, 6, 5]])
M

M.echelon_form()

EXAMPLE 3  System with Fewer Equations than Variables

$3x + 7y + 6z = 26$
$x + 2y +  z = 8$

M = matrix([[3, 7, 6, 26], [1, 2, 1, 8]])
M

M.swap_rows(0, 1)
M

M[1] = -3*M[0] + M[1]
M

var('x y z')

y = 2 - 3*z
x = 8 - z - 2*y

(x, y, z)