We are given the system of equations:

10x + 6y = 14 \tag{1} 4y + 3z = 5 \tag{2} 20x + 7z = 11 \tag{3}

We need to find the value of $3x + y + z$.

Step 1: Solve for one variable in terms of others

Start with equation (1) and simplify it:

$10x + 6y = 14$ Divide the whole equation by 2:

5x + 3y = 7 \tag{4}

Now, from equation (4), solve for $y$ in terms of $x$:

$3y = 7 - 5x$ y = \frac{7 - 5x}{3} \tag{5}

Step 2: Substitute $y$ in equation (2)

Now, substitute equation (5) into equation (2):

$4y + 3z = 5$ Substitute for $y$:

$4\left(\frac{7 - 5x}{3}\right) + 3z = 5$ Simplify:

$\frac{4(7 - 5x)}{3} + 3z = 5$ $\frac{28 - 20x}{3} + 3z = 5$ Multiply through by 3 to eliminate the fraction:

$28 - 20x + 9z = 15$ Simplify:

-20x + 9z = -13 \tag{6}

Step 3: Solve for $z$ in terms of $x$

Now, we have a simplified system:

-20x + 9z = -13 \tag{6} 20x + 7z = 11 \tag{3}

To solve for $z$, let's eliminate $x$ by adding the two equations. Multiply equation (6) by 1 and equation (3) by 1:

-20x + 9z = -13 \tag{6} 20x + 7z = 11 \tag{3}

Add them together:

$(0) + 16z = -2$ $16z = -2$ $z = -\frac{2}{16} = -\frac{1}{8}$

Step 4: Substitute $z$ into equation (5) and find $y$

Now that we know $z = -\frac{1}{8}$, substitute this into equation (5) to find $y$:

$y = \frac{7 - 5x}{3}$

From equation (3), solve for $x$.