Find XY if Y is in between X and Z with the given XY = 2x + 1, YZ = 6x, and XZ = 81

Given the problem where \( Y \) is between \( X \) and \( Z \), with \( XY = 2x + 1 \), \( YZ = 6x \), and \( XZ = 81 \), we can use the relationship:

\[
XY + YZ = XZ
\]

Substitute the given values:

\[
(2x + 1) + 6x = 81
\]

Combine like terms:

\[
8x + 1 = 81
\]

Subtract 1 from both sides:

\[
8x = 80
\]

Divide by 8:

\[
x = 10
\]

Now, substitute \( x = 10 \) back into the expression for \( XY \):

\[
XY = 2x + 1 = 2(10) + 1 = 20 + 1 = 21
\]

**Answer:**
The length of \( XY \) is 21.

Would you like more details or have any questions?

Here are five related questions:
1. What is the length of \( YZ \) with the calculated value of \( x \)?
2. If \( XZ \) were a different value, how would you adjust the calculations?
3. What does it mean geometrically for \( Y \) to be between \( X \) and \( Z \)?
4. How do you solve for \( x \) when given equations involving segments?
5. What would happen if \( Y \) was not exactly between \( X \) and \( Z \) in terms of segment addition?

**Tip:** Always double-check the equation setup to ensure that all segments are accounted for correctly when applying the segment addition postulate.

Learn how to find the length of segment XY using the Segment Addition Postulate when Y is between X and Z. Solve the linear equation 8x + 1 = 81 step-by-step. Suitable for high school students studying segment addition in geometry.

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