Find all solutions to the system of linear equations: C18, Page 42 \[x_{1}+2x_{2}-4x_{3}-x_{4} = 32\] \[x_{1}+3x_{2}-7x_{3}-x_{5} = 33\] \[x_{1}+2x_{3}-2x_{4}+3x_{5} = 22\]

library(matlib)

A <- matrix(c(1, 1, 1, 2, 3, 0, -4, -7, 2, -1, 0, -2, 0, -1, 3), 3, 5)
b <- c(32, 33, 22)

showEqn(A, b) # show Matrices (A, b) as Linear Equations
## 1*x1 + 2*x2 - 4*x3 - 1*x4 + 0*x5  =  32 
## 1*x1 + 3*x2 - 7*x3 + 0*x4 - 1*x5  =  33 
## 1*x1 + 0*x2 + 2*x3 - 2*x4 + 3*x5  =  22
gaussianElimination(A, b) # row reduction used for solving systems of linear equations
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    0    2    0    5    6
## [2,]    0    1   -3    0   -2    9
## [3,]    0    0    0    1    1   -8
Solve(A, b) # solve and Display Solutions for Systems of Linear Simultaneous Equations
## x1   + 2*x3   + 5*x5  =   6 
##   x2 - 3*x3   - 2*x5  =   9 
##            x4   + x5  =  -8