We are given a triangle $VWX$ with sides $VW = 3y + 3$, $VX = 4y$, and $WX = y + 1$. The perimeter of the triangle is 44 units, and we are asked to find the length of side $VW$.

The perimeter of the triangle is the sum of the lengths of all its sides. Therefore, the equation for the perimeter is:

$VW + VX + WX = 44$

Substituting the expressions for the sides:

$(3y + 3) + 4y + (y + 1) = 44$

Now let's solve this equation step by step.

1. Combine like terms:

$3y + 3 + 4y + y + 1 = 44$ $8y + 4 = 44$

2. Subtract 4 from both sides:

$8y = 40$

3. Solve for $y$:

$y = \frac{40}{8} = 5$

Now that we know $y = 5$, substitute this value back into the expression for $VW = 3y + 3$:

$VW = 3(5) + 3 = 15 + 3 = 18$

Thus, the length of side $VW$ is $\boxed{18}$.

Do you want more details on this? Feel free to ask!

Here are some related questions:

1. What would be the length of side $VX$ after solving for $y$?
2. Can you calculate the length of side $WX$ after solving for $y$?
3. How does the formula for the perimeter change for different triangles?
4. What are some other methods to solve for the perimeter of a triangle?
5. How can we verify that our solution for $y$ is correct?

Tip: Always double-check your work by substituting the value you found back into the original equation.