Final answer:
To find the length of ZY when Y is the midpoint between X and Z, set the expressions for XY and YZ equal, solve for x, then substitute the value back into the expression for either segment to find the length of ZY.
Step-by-step explanation:
The problem involves determining the length of ZY in a line segment XZ, where Y is the midpoint. By recognizing that in a midpoint scenario, XY is equal to YZ, the equation 24x + 7 = 2x + 26 is established. Solving for x yields x = 1. Substituting this value back into either XY or YZ gives the length of ZY. Choosing XY, we find ZY = 24(1) + 7 = 31. The value of x is validated by the provided reference, ensuring accuracy in the solution.
This problem showcases the application of the midpoint theorem and algebraic manipulation to determine segment lengths in a geometric context, offering a comprehensive understanding of the relationship between the given line segments.