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

We are given that

The system of equations is:

$$x + y + az = b \quad \cdots (1)$$

$$2x + 5y + 2z = 6 \quad \cdots (2)$$

$$x + 2y + 3z = 3 \quad \cdots (3)$$

The system has infinitely many solutions, and we need to find $$2a + 3b$$.

To begin,

For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero. The coefficient matrix is:

$$D = \begin{vmatrix} 1 & 1 & a \\ 2 & 5 & 2 \\ 1 & 2 & 3 \end{vmatrix}$$

Expanding along the first row:

$$D = 1(5 \times 3 - 2 \times 2) - 1(2 \times 3 - 2 \times 1) + a(2 \times 2 - 5 \times 1)$$

$$= 1(15 - 4) - 1(6 - 2) + a(4 - 5) = 11 - 4 - a = 7 - a$$

Setting $$D = 0$$: $$a = 7$$.

Next,

For infinitely many solutions, all augmented determinants must also be zero. Replace the third column (the z-coefficient column) with the constants column:

$$D_3 = \begin{vmatrix} 1 & 1 & b \\ 2 & 5 & 6 \\ 1 & 2 & 3 \end{vmatrix} = 1(15 - 12) - 1(6 - 6) + b(4 - 5) = 3 - 0 - b = 3 - b$$

Setting $$D_3 = 0$$: $$b = 3$$.

From here,

Replace the first column with the constants:

$$D_1 = \begin{vmatrix} 3 & 1 & 7 \\ 6 & 5 & 2 \\ 3 & 2 & 3 \end{vmatrix} = 3(15 - 4) - 1(18 - 6) + 7(12 - 15) = 33 - 12 - 21 = 0 \; \checkmark$$

Continuing,

$$2a + 3b = 2(7) + 3(3) = 14 + 9 = 23$$

The correct answer is Option (3): 23.