#Read data
y <- matrix(c(10,15,17,22,23,NA),2,3,byrow=TRUE)

#Parameter vector is theta=(mu,alpha1,beta1,beta2) with
#alpha2 = -alpha1 and beta3 = -beta1-beta2.
X <- cbind(rep(1,5),c(1,-1,1,-1,1),c(1,1,0,0,-1),c(0,0,1,1,-1))

#Least-Square-Estimator
theta.hat <- solve(t(X) %*% X) %*% t(X) %*%  matrix(c(y)[-6],5,1)
row.names(theta.hat) <- c("mu","alpha","beta1","beta2")
t(theta.hat)