library(matlib)
A <- matrix(data = c(2, -1, 5,
                     -4, 2, -10), nrow = 2, ncol = 3, byrow = T)

t1 <- matrix(data = c(2, 3), nrow = 2, ncol = 1, byrow = F)


B <- cbind(A, t1)

# Row-reduced form of B:
echelon(B)
##      [,1] [,2] [,3] [,4]
## [1,]    1 -0.5  2.5    0
## [2,]    0  0.0  0.0    1

The above system is inconsistent.

t2 <- matrix(data = c(4, -8), nrow = 2, ncol = 1, byrow = F)
C <- cbind(A, t2)

# Row-reduced form of B:
echelon(C)
##      [,1] [,2] [,3] [,4]
## [1,]    1 -0.5  2.5    2
## [2,]    0  0.0  0.0    0

From the above we see that the solution is of the form v1 + v2 + v3 where

\[ {v1} = \left[\begin{array} {rrr} 2\\ 0\\ 0 \end{array}\right] \]

\[ {v2} = k1\left[\begin{array} {rrr} 0.5\\ 1\\ 0 \end{array}\right] \]

\[ {v3} = k2\left[\begin{array} {rrr} -2.5\\ 0\\ 1 \end{array}\right] \]

and k1, k2 are independent variables.