If the system of linear equations
$$3x + y + \beta z = 3$$
$$2x + \alpha y - z = -3$$
$$x + 2y + z = 4$$
has infinitely many solutions, then the value of $$22\beta - 9\alpha$$ is:

For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix ($$D$$) and the determinants associated with each variable ($$D_x, D_y, D_z$$) must all equal zero.

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Step 1: Compute the main determinant $$D$$ and set it to zero

$$D = \begin{vmatrix} 3 & 1 & \beta \\ 2 & \alpha & -1 \\ 1 & 2 & 1 \end{vmatrix} = 0$$

Expanding the determinant along the first row:

$$D = 3(\alpha(1) - (-1)(2)) - 1(2(1) - (-1)(1)) + \beta(2(2) - \alpha(1)) = 0$$

$$D = 3(\alpha + 2) - 1(3) + \beta(4 - \alpha) = 0$$

$$3\alpha + 6 - 3 + 4\beta - \alpha\beta = 0 \implies 3\alpha + 4\beta - \alpha\beta + 3 = 0$$

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Step 2: Compute the determinant $$D_y$$ and set it to zero

To simplify calculations, we substitute the constant column into the second column to find $$D_y$$:

$$D_y = \begin{vmatrix} 3 & 3 & \beta \\ 2 & -3 & -1 \\ 1 & 4 & 1 \end{vmatrix} = 0$$

Expanding the determinant along the first row:

$$D_y = 3(-3(1) - (-1)(4)) - 3(2(1) - (-1)(1)) + \beta(2(4) - (-3)(1)) = 0$$

$$D_y = 3(-3 + 4) - 3(2 + 1) + \beta(8 + 3) = 0$$

$$3(1) - 3(3) + 11\beta = 0 \implies 3 - 9 + 11\beta = 0$$

$$11\beta - 6 = 0 \implies \beta = \frac{6}{11}$$

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Step 3: Solve for $$\alpha$$

Substituting the value of $$\beta = \frac{6}{11}$$ back into the equation for $$D = 0$$:

$$3\alpha + 4\left(\frac{6}{11}\right) - \alpha\left(\frac{6}{11}\right) + 3 = 0$$

$$3\alpha - \frac{6}{11}\alpha + \frac{24}{11} + 3 = 0$$

$$\frac{27}{11}\alpha + \frac{57}{11} = 0 \implies 27\alpha = -57 \implies \alpha = -\frac{57}{27} = -\frac{19}{9}$$

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Step 4: Evaluate the required expression

Now, substitute the values of $$\alpha = -\frac{19}{9}$$ and $$\beta = \frac{6}{11}$$ into the given expression:

$$22\beta - 9\alpha = 22\left(\frac{6}{11}\right) - 9\left(-\frac{19}{9}\right)$$

$$22\beta - 9\alpha = 2(6) + 19 = 12 + 19 = 31$$

Therefore, the value of $$22\beta - 9\alpha$$ is equal to 31.