In obtaining the solution of the system of equations x + y + z = 7 , x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer's rule, the value of y is obtained by dividing D by D₂ , where D = vmatrix 1 & 1 & 1 1 & 2 & 3 1 & 3 & 4 vmatrix . What is the value of the determinant D₂ ?

Step-by-step solution

The given system of linear equations is: x + y + z = 7 x + 2y + 3z = 16 x + 3y + 4z = 22 By Cramer's rule, the value of y is given by y = D₂ D , where D₂ is the determinant obtained by replacing the second column of the coefficient matrix D with the column of constant terms. D₂ = vmatrix 1 & 7 & 1 1 & 16 & 3 1 & 22 & 4 vmatrix Expanding the determinant along the first row: D₂ = 1(16 4 - 3 22) - 7(1 4 - 3 1) + 1(1 22 - 16 1) D₂ = 1(64 - 66) - 7(4 - 3) + 1(22 - 16) D₂ = 1(-2) - 7(1) + 1(6) D₂ = -2 - 7 + 6 D₂ = -3 Ans