Additive property of length 7RA

Additive property of length 7RA
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If EF = 2x, FG = 8, and EG = 4x, what is ER?
E
F
G
2x
4x
Simplify your answer and write it as a proper fraction, mixed number, or integer.

The Correct Answer and Explanation is :

To solve for ( ER ) given the segments ( EF = 2x ), ( FG = 8 ), and ( EG = 4x ), we can utilize the additive property of length, which states that the total length of a line segment is equal to the sum of its parts.

Step 1: Understand the Relationships

We are given three segments along a line:

1. ( EF = 2x )
2. ( FG = 8 )
3. ( EG = 4x )

According to the additive property, we can write the relationship as:
[
EG = EF + FG
]
Substituting the known expressions into this equation gives:
[
4x = 2x + 8
]

Step 2: Solve for ( x )

Now we can solve for ( x ):

1. Subtract ( 2x ) from both sides:
   [
   4x – 2x = 8
   ]
   [
   2x = 8
   ]
2. Divide both sides by 2:
   [
   x = 4
   ]

Step 3: Find the Lengths of Each Segment

Now that we have the value of ( x ), we can find the lengths of each segment:

1. Length of ( EF ):
   [
   EF = 2x = 2(4) = 8
   ]
2. Length of ( FG ):
   [
   FG = 8
   ]
3. Length of ( EG ):
   [
   EG = 4x = 4(4) = 16
   ]

Step 4: Find ( ER )

Now, to find ( ER ), we notice that ( ER ) is actually equal to ( EG ), since ( E ) and ( G ) are the endpoints of the entire segment. Thus,
[
ER = EG = 16
]

Conclusion

Therefore, the value of ( ER ) is ( \boxed{16} ).

Explanation

The additive property of length helps us understand how segments relate to each other on a number line. By setting up an equation based on these segments and solving for ( x ), we can easily calculate the lengths of the individual segments and ultimately find the desired total length. This approach exemplifies how algebra can simplify geometric problems, making it a valuable tool in various mathematical applications.