The problem is asking for the length of side $XY$, given that the perimeter of the rectangle is 86 units.

Step-by-step solution:

1. Perimeter formula for a rectangle:
   The perimeter of a rectangle is given by
   $P = 2 \times (\text{length} + \text{width})$ Here, the length is $3z + 2$ (side $VW$) and the width is $2z + 1$ (side $YV$).

2. Substitute the given perimeter:
   From the problem, $P = 86$. Substituting into the formula:
   $86 = 2 \times \left((3z + 2) + (2z + 1)\right)$

3. Simplify inside the parentheses:
   $(3z + 2) + (2z + 1) = 5z + 3$ Thus,
   $86 = 2 \times (5z + 3)$

4. Divide by 2 to isolate the term:
   $43 = 5z + 3$

5. Solve for $z$:
   Subtract 3 from both sides:
   $40 = 5z$
   Divide by 5:
   $z = 8$

6. Find the length of $XY$:
   Side $XY$ corresponds to $2z + 1$. Substitute $z = 8$:
   $XY = 2(8) + 1 = 16 + 1 = 17$

Final Answer:

$\boxed{17 \, \text{units}}$

Would you like further clarifications or detailed steps?

Related Questions:

1. What would the width $VW$ of the rectangle be if $z = 8$?
2. How does the formula for perimeter change if the rectangle becomes a square?
3. If the perimeter were 100 instead of 86, what would $XY$ become?
4. How does solving for $z$ change if the length and width are given as different expressions?
5. What would the dimensions of the rectangle be in terms of $z$ before solving?

Tip:

Always double-check your algebraic simplifications and substitution steps to avoid small errors!