Question
Given the rectangle ABCD as shown, find the value of x if BE = 4x + 3 and EC = 10x - 15.
Answers
GPT-4o mini
To solve for \( x \) in the triangle, we will set the two expressions for \( BE \) and \( EC \) equal to each other since \( E \) is a point on \( BC \) (hence, the segments \( BE \) and \( EC \) must sum to the total length of \( BC \)).
Given:
- \( BE = 4x + 3 \)
- \( EC = 10x - 15 \)
Since point \( E \) divides line segment \( BC \), we know that:
\[
BC = BE + EC
\]
Equating the two segments:
\[
BE + EC = BC
\]
However, our goal is to simply set \( BE \) equal to \( EC \):
\[
BE = EC
\]
Thus:
\[
4x + 3 = 10x - 15
\]
Next, we will solve the equation for \( x \).
1. Subtract \( 4x \) from both sides:
\[
3 = 10x - 4x - 15
\]
\[
3 = 6x - 15
\]
2. Next, add \( 15 \) to both sides:
\[
3 + 15 = 6x
\]
\[
18 = 6x
\]
3. Lastly, divide both sides by \( 6 \):
\[
x = \frac{18}{6} = 3
\]
Thus, the value of \( x \) is \( \boxed{3} \).
Given:
- \( BE = 4x + 3 \)
- \( EC = 10x - 15 \)
Since point \( E \) divides line segment \( BC \), we know that:
\[
BC = BE + EC
\]
Equating the two segments:
\[
BE + EC = BC
\]
However, our goal is to simply set \( BE \) equal to \( EC \):
\[
BE = EC
\]
Thus:
\[
4x + 3 = 10x - 15
\]
Next, we will solve the equation for \( x \).
1. Subtract \( 4x \) from both sides:
\[
3 = 10x - 4x - 15
\]
\[
3 = 6x - 15
\]
2. Next, add \( 15 \) to both sides:
\[
3 + 15 = 6x
\]
\[
18 = 6x
\]
3. Lastly, divide both sides by \( 6 \):
\[
x = \frac{18}{6} = 3
\]
Thus, the value of \( x \) is \( \boxed{3} \).