A fair six-sided die is rolled three times, and the outcomes are recorded as a , b , and c , respectively. Let p be the probability that the system of linear equations array l x + y + z = a x + 2y + 3z = b 2x + 3y + 4z = c array has infinitely many solutions. Then the value of 72p is

Step-by-step solution

The determinant of the coefficient matrix is: = vmatrix 1 & 1 & 1 1 & 2 & 3 2 & 3 & 4 vmatrix = 1(8 - 9) - 1(4 - 6) + 1(3 - 4) = -1 + 2 - 1 = 0 Since = 0 , the system will have infinitely many solutions if the equations are linearly dependent. Observing the coefficients, we see that the third equation's left-hand side is exactly the sum of the first two: (x + y + z) + (x + 2y + 3z) = 2x + 3y + 4z For the system to be consistent (and thus have infinitely many solutions), the constants must satisfy the same relations