To find $CE$, let's analyze the given information from the image:

Line segment details:

• $BC = 3x + 47$
• $CD = 10$
• The total length of $BE$ is composed of two parts:
  • $BD = 27 + x$
  • $DE = x + 26$

We know that $BE$ can be expressed in terms of these two segments: $BE = BD + DE = (27 + x) + (x + 26)$ Simplifying this: $BE = 27 + x + x + 26 = 2x + 53$

But we also know that $BE$ is equal to the sum of $BC$, $CD$, and $DE$. So: $BE = BC + CD + DE = (3x + 47) + 10 + (x + 26)$ Simplifying the right-hand side: $BE = 3x + 47 + 10 + x + 26 = 4x + 83$

Now, equate the two expressions for $BE$: $2x + 53 = 4x + 83$ Solve for $x$: $53 - 83 = 4x - 2x$ $-30 = 2x$ $x = -15$

Finding $CE$:

Now that we have $x = -15$, we can find $CE$. We know that: $CE = CD + DE = 10 + (x + 26)$ Substitute $x = -15$ into this expression: $CE = 10 + (-15 + 26) = 10 + 11 = 21$

So, the length of $CE$ is 21 units.

Would you like more details or further explanations on any part of this solution?

Here are 5 related questions you might want to explore:

1. How would the answer change if $x$ had a different value?
2. Can you prove the calculation for $BE$ using a different method?
3. What is the length of segment $BC$ when $x = -15$?
4. How would you calculate the total length of $BE$?
5. What geometric principles can you apply to similar problems?

Tip: When solving linear equations involving segments, breaking down the total distance into smaller known parts can make the calculations clearer.