### 5.61.6 Linear system solving: linsolve

linsolve is used to solve a system of linear equations.
linsolve takes its arguments in two different ways.

- It can take two arguments, the first is a list of equations or expressions (in that case the convention is that the equation is expression = 0), and a list of variable names.
  linsolve returns the solution of the system in a list.
  Input:
  linsolve([2*x+y+z=1,x+y+2*z=1,x+2*y+z=4],[x,y,z])
  Output :
  [1/-2,5/2,1/-2]
  Which means that

  x=−

  1
  2

  , y=

  5
  2

  , z=−

  1
  2

  is the solution of the system :

  ⎧
  ⎪
  ⎨
  ⎪
  ⎩

  2x+y+z | =1
  x+y+2z | =1
  x+2y+z | =4

- It can take two arguments, the matrix of coefficients of a system and values of the right hand side in the form of a list.
  Input:
  linsolve ([[2,1,1], [1,1,2], [1,2,1]], [1,1,4])
  Output:
  [-1/2,5/2,-1/2]
- It can take four arguments; the matrices P, L, U from the lu decomposition and the values of the right hand side in the form of a list. This is useful when you have several systems of equations which only differ on their right hand side.
  Input:
  p,l,u:=lu([[2,1,1],[1,1,2],[1,2,1]])
  linsolve(p,l,u,[1,1,4])
  Output:
  [-1/2,5/2,-1/2]

If the Step by step option is checked in the general configuration, a window will also pop up showing:

     Matrix [[1,1,2, -1], [0,1, -1, -3], [0, -1, -3,1]]
     Row operation L2 <- (1) * L1- (1) * L2
     Matrix [[1,0,3,2], [0,1, -1, -3], [0, -1, -3,1]]
     Row operation L2 <- (1) * L3 - (- 1) * L2
     Matrix [[1,0,3,2], [0,1, -1, -3], [0,0, -4, -2]]
     Reducing column 3 using pivot -4 at row 3
     Matrix [[1,0,3,2], [0,1, -1, -3], [0,0, -4, -2]]
     Row operation L3 <- (-4) * L1- (3) * L3
     Matrix [[-4,0,0, -2], [0,1, -1, -3], [0,0, -4, -2]]
     Row operation L3 <- (-4) * L2 - (- 1) * L3
     End reduction [[-4,0,0, -2], [0, -4,0,10], [0,0, -4, -2]]

The linsolve command also solves systems with coefficients in ℤ/nℤ.
Input:

linsolve([2*x+y+z-1,x+y+2*z-1,x+2*y+z-4]%3,[x,y,z])

Output:

[1 % 3,1 % 3,1 % 3]