If the system of linear equations x_1 + 2x_2 | vedclass

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(D) The given system of equations can be written in matrix form $AX = B$,where $A = \begin{bmatrix} 1 & 2 & 3 \ 1 & 3 & 5 \ 2 & 5 & a \end{bmatrix}$

Vedclass · Accepted Answer

$a = 8, b = 15$

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(D) The given system of equations can be written in matrix form $AX = B$,where $A = \begin{bmatrix} 1 & 2 & 3 \ 1 & 3 & 5 \ 2 & 5 & a \end{bmatrix}$ and $B = \begin{bmatrix} 6 \ 9 \ b \end{bmatrix}$.For the system to have infinitely many solutions,the determinant of the augmented matrix must be zero,and the rank of the augmented matrix must be less than the number of variables.First,calculate the determinant of $A$: $|A| = 1(3a - 25) - 2(a - 10) + 3(5 - 6) = 3a - 25 - 2a + 20 - 3 = a - 8$.For infinite solutions,$|A| = 0$,which implies $a = 8$.Now,substitute $a = 8$ into the augmented matrix $[A|B] = \begin{bmatrix} 1 & 2 & 3 & | & 6 \ 1 & 3 & 5 & | & 9 \ 2 & 5 & 8 & | & b \end{bmatrix}$.Perform row operations: $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - 2R_1$:$\begin{bmatrix} 1 & 2 & 3 & | & 6 \ 0 & 1 & 2 & | & 3 \ 0 & 1 & 2 & | & b - 12 \end{bmatrix}$.For the system to be consistent with infinite solutions,the last two rows must be identical,so $b - 12 = 3$,which gives $b = 15$.Thus,$a = 8$ and $b = 15$.

If the system of linear equations $x_1 + 2x_2 + 3x_3 = 6$,$x_1 + 3x_2 + 5x_3 = 9$,and $2x_1 + 5x_2 + ax_3 = b$ is consistent and has an infinite number of solutions,then:

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