lineq <- function(a){
r <- rep(NA,14)
v <- a[1]
O1 <- a[2]
O2 <- a[3]
O0 <- v*O1 + (1-v)*O2 

r <- rep(-1,14)

kount <- 0

while (any(r < 0)) {
kount <- kount + 1 
x <- runif(7)
U0 <- x[1]
U1 <- x[2]
U2 <- x[3]
q0 <- x[4]
q1 <- x[5]
q2 <- x[6]
p <- x[7]

r[1] <- U0
r[2] <- U1
r[3] <- U2
r[4] <- U0 - (U1 + (O1 - O0)*q1)
r[5] <- U0 - (U2 + (O2 - O0)*q2)
r[6] <- U1 - (U0 + (O0 - O1)*q0)
r[7] <- U1 - (U2 + (O2 - O1)*q2)
r[8] <- U2 - (U0 + (O0 - O2)*q0)
r[9] <- U2 - (U1 + (O1 - O2)*q1)
r[10] <- p
r[11] <- 1-p
r[12] <- q0
r[13] <- q1
r[14] <- q2
}

list(x0=x, kount=kount, r=sign(r)) 
}

a <- c(0.9,10,20)
ans <- lineq(a)
ans$x0   #  interior point