In rectangle BCDA, point E lies on line segment BC, which means that the lengths of segments BE and EC must add up to the length of segment BC.
We have the following expressions for the lengths of segments:
- \( BE = 2x + 9 \)
- \( EC = 12x - 11 \)
Since \( E \) is a point on \( BC \), we can set up the equation:
\[ BE + EC = BC \]
However, we do not have the length of BC directly. Instead, we can use their relationship as follows:
\[ 2x + 9 + 12x - 11 = BC \]
This simplifies to:
\[ (2x + 12x) + (9 - 11) = 14x - 2 \]
Now, to find the value of \( x \), we know that since point E is somewhere on line BC, we can say that the lengths BE and EC together cover the length of BC, effectively making them equal in total length.
Thus, we have:
\[ 2x + 9 + 12x - 11 = 0 \text{ (This is incorrect since it shouldn't equate to zero, but instead checking equality)} \]
Thus, we actually seek a practical resolution – setting BE equal to some hypothetical value (which is BC), requiring both segments to cover the length equally.
To equate for \( B \) and \( C \) simply:
\[ 2x + 9 = 12x - 11 \]
We can now collect like terms and solve for \( x \):
- Move all \( x \) terms to one side:
\[ 2x + 9 + 11 = 12x \]
- Simplifying gives:
\[ 20 = 12x - 2x \]
- Thus,
\[ 20 = 10x \]
- Dividing by \( 10 \):
\[ x = 2 \]
Thus, the value of \( x \) is:
\[ \boxed{2} \]
To check, substituting \( x = 2 \) back into our segment lengths:
- \( BE = 2(2) + 9 = 4 + 9 = 13 \)
- \( EC = 12(2) - 11 = 24 - 11 = 13 \)
We confirm that \( BE + EC = 13 + 13 = 26 \) matches \( BC \), holding the values consistent.
Thus, \( x \) is correct:
\[ \boxed{2} \]