If the system of equationsx+y+az=b2x+5y+2z=6x | vedclass

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(A) For a system of linear equations to have infinitely many solutions,the determinant of the coefficient matrix $\Delta$ must be $0$,and the determin

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$23$

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(A) For a system of linear equations to have infinitely many solutions,the determinant of the coefficient matrix $\Delta$ must be $0$,and the determinants $\Delta_x, \Delta_y, \Delta_z$ must also be $0$.First,calculate $\Delta$:$\Delta = \begin{vmatrix} 1 & 1 & a \ 2 & 5 & 2 \ 1 & 2 & 3 \end{vmatrix} = 1(15-4) - 1(6-2) + a(4-5) = 11 - 4 - a = 7 - a$.Setting $\Delta = 0$,we get $7 - a = 0$,so $a = 7$.Next,for infinitely many solutions,$\Delta_x = 0$ (or $\Delta_y = 0$ or $\Delta_z = 0$):$\Delta_x = \begin{vmatrix} b & 1 & 7 \ 6 & 5 & 2 \ 3 & 2 & 3 \end{vmatrix} = b(15-4) - 1(18-6) + 7(12-15) = 11b - 12 - 21 = 11b - 33$.Setting $\Delta_x = 0$,we get $11b = 33$,so $b = 3$.Finally,calculate $2a + 3b$:$2a + 3b = 2(7) + 3(3) = 14 + 9 = 23$.

If the system of equations
$x+y+az=b$
$2x+5y+2z=6$
$x+2y+3z=3$
has infinitely many solutions,then $2a+3b$ is equal to $...........$.

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If the system of equations$x+y+az=b$$2x+5y+2z=6$$x+2y+3z=3$has infinitely many solutions,then $2a+3b$ is equal to $...........$.