If the system of linear equations: $\begin{aligned} x + y + 2z &= 6 \\ 2x + 3y + az &= a + 1 \\ -x - 3y + bz &= 2b \end{aligned}$ where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to:

Step 1: Write the augmented matrix

To solve a system of linear equations, we first write its augmented matrix. This matrix combines the coefficients of the variables and the constants on the right-hand side of the equations. Each row represents an equation, and each column (before the vertical bar) corresponds to a variable.

Step 2: Perform row operations to simplify the matrix

We perform elementary row operations to transform the matrix into an echelon form. First, we replace $R_2$ with $R_2 - 2R_1$ and $R_3$ with $R_3 + R_1$. Then, we replace $R_3$ with $R_3 + 2R_2$. These operations help us eliminate variables and simplify the system.

Step 3: Apply condition for infinitely many solutions

For a system of linear equations to have infinitely many solutions, the last row of the echelon form of the augmented matrix must be entirely zeros. This means both the coefficient of $z$ and the constant term in the last row must be zero.

Step 4: Solve the system of equations for $a$ and $b$

We now have a system of two linear equations with two variables, $a$ and $b$: $a+b=10$ and $a+2b=17$. Subtracting the first equation from the second gives $b=7$. Substituting $b=7$ into the first equation gives $a+7=10$, so $a=3$.

Step 5: Calculate $7a + 3b$

Finally, we substitute the values of $a=3$ and $b=7$ into the expression $7a + 3b$ to find the required value.

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