The ordered pair (a, b),for which the system | vedclass

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(C) For a system of linear equations to have no solution,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of the Cramer

Vedclass · Accepted Answer

$\left(-3, -\frac{1}{3}\right)$

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(C) For a system of linear equations to have no solution,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of the Cramer's rule determinants $(\Delta_x, \Delta_y, \Delta_z)$ must be non-zero.First,calculate the determinant of the coefficient matrix $\Delta$:$\Delta = \begin{vmatrix} 3 & -2 & 1 \ 5 & -8 & 9 \ 2 & 1 & a \end{vmatrix} = 3(-8a - 9) + 2(5a - 18) + 1(5 - (-16))$$\Delta = -24a - 27 + 10a - 36 + 21 = -14a - 42$Setting $\Delta = 0$,we get $-14a = 42$,so $a = -3$.Now,substitute $a = -3$ into the system and check for consistency using the augmented matrix or Cramer's rule. For no solution,the system must be inconsistent.Consider the determinant $\Delta_z = \begin{vmatrix} 3 & -2 & b \ 5 & -8 & 3 \ 2 & 1 & -1 \end{vmatrix} = 3(8 - 3) + 2(-5 - 6) + b(5 - (-16))$$\Delta_z = 3(5) + 2(-11) + b(21) = 15 - 22 + 21b = 21b - 7$.For no solution,we require $\Delta = 0$ and $\Delta_z 
eq 0$ (or other $\Delta_i 
eq 0$).However,checking the consistency of the equations $3x - 2y + z = b$,$5x - 8y + 9z = 3$,and $2x + y - 3z = -1$:Adding the equations or using row reduction,we find that for the system to be inconsistent,$b$ must be such that the constants do not satisfy the linear dependence of the rows. Solving the system leads to $b = -\frac{1}{3}$.

The ordered pair $(a, b)$,for which the system of linear equations $3x - 2y + z = b$,$5x - 8y + 9z = 3$,and $2x + y + az = -1$ has no solution,is

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