Question (d): Infinite domain III (20 pts) Suppose a, b and c are integers (Z), while x, y and z are non-zero reals (R 6=0). If the following relations hold for these numbers: a = x · y x + y b = x · z x + z c = y · z y + z prove that x, (not necessarily y or z) is a rational number (Q). Some hints: (i) How can you prove that a certain number is rational? (ii) Can a, b or c be zero? Why? You would have to mention why if you need to show us that they are non-zero! (iii) If you see yourself doing a lot of algebra... keep going! Find a way to get x alone and integers on the other side of the equality.

Asystem of linear equations with more equations than unknowns is sometimes called an overdetermined system. can such a system be consistent? illustrate your answer with a specific system of three equations in two unknowns. choose the correct answer below. a. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 b. no, overdetermined systems cannot be consistent because there are fewer free variables than equations. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 c. yes, overdetermined systems can be consistent. for example, the system of equations below is consistent because it has the solution nothing. (type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 d. no, overdetermined systems cannot be consistent because there are no free variables. for example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24