Isosceles triangle from symbolic perimeter:\nThe base of an isosceles triangle is B = 2x + 2y + 4z and the perimeter is P = 4x + 2y + 6z (expressions in the same units). Find the length of each equal side.

Introduction / Context:
For an isosceles triangle with base B and equal sides s, perimeter P satisfies P = B + 2s. Rearranging gives s = (P − B)/2. Here B and P are algebraic expressions in x, y, z that represent like units (e.g., centimetres).

Given Data / Assumptions:

• Base B = 2x + 2y + 4z.
• Perimeter P = 4x + 2y + 6z.
• All symbols represent nonnegative measures in the same unit.

Concept / Approach:
Apply s = (P − B)/2 algebraically and simplify the expression; this yields the length of each congruent side in terms of x and z only, since the y-terms cancel here.

Step-by-Step Solution:

Verification / Alternative check:
Perimeter re-formed: B + 2s = (2x + 2y + 4z) + 2(x + z) = 4x + 2y + 6z = P, consistent.

Why Other Options Are Wrong:
Options involving y do not reflect the cancellation; 2(x + y) doubles the wrong combination; x + y is not supported by P − B reduction.

Common Pitfalls:
Forgetting the factor 1/2 in s = (P − B)/2; algebraic sign mistakes.

Final Answer:
x+z