You want an example for solving least absolute deviation by linear programming. I will show you an simple implementation in R. Quantile regression is a generalization of least absolute deviation, which is the case of the quantile 0.5, so I will show a solution for quantile regression. Then you can check the results with the R quantreg package:

    rq_LP  <-  function(x, Y, r=0.5, intercept=TRUE) {
        require("lpSolve")
        if (intercept) X  <-  cbind(1, x) else X <-  cbind(x)
        N   <-  length(Y)
        n  <-  nrow(X)
        stopifnot(n == N)
        p  <-  ncol(X)
        c  <-  c(rep(r, n), rep(1-r, n), rep(0, 2*p))  
                 # cost coefficient vector
        A  <- cbind(diag(n), -diag(n), X, -X)
        res  <-  lp("min", c, A, "=", Y, compute.sens=1)
            ### Desempaquetar los coefs:
        sol <- res$solution
        coef1  <-  sol[(2*n+1):(2*n+2*p)]
        coef <- numeric(length=p)
        for (i in seq(along=coef)) {
             coef[i] <- (if(coef1[i]<=0)-1 else +1) *  
                  max(coef1[i], coef1[i+p])
        }
        return(coef)
        }

Then we use it in a simple example:

    library(robustbase)
    data(starsCYG)
    Y  <- starsCYG[, 2]
    x  <- starsCYG[, 1]
    rq_LP(x, Y)
    [1]  8.1492045 -0.6931818

then you yourself can do the check with quantreg.

I've seen a comment saying that I can't treat the absolute value the way I do, but the logic is as follows.

We have e_i = | y_i - a - b x_i |

Since we are minimising the sum of e_i, at the optimal point, this is equivalent to e_i >= | y_i - a -b x_i |. From there, it is separated into positive and negative cases.