help please. Thank You very kindly. addition postulate?

Question

Suppose M is between L and N. Use the segment Addition Postulate to solve for the variable. Then find the lengths of line: LM MN and LN

LM = 1/2z + 2
MN = 3z + 3/2
LN = 5z + 2

Answer 1

Using the addition postulate (which states one part of a line + another part of the line = the whole line), you get LM+MN=LN. This comes out to be:

1/2z+2+3z+3/2=5z+2

Combine like terms

-3/2z=-3/2

Divide both sides by -3/2

z=1

Therefore, LM=2.5, MN=4.5, and LN=7

Answer 2

L-------M-------N

Point M is a point on segment LN, i.e. M is between L and N, if and only if

LM + MN = LN

Substitute values of LM, MN and LN from the question:
1/2z + 2 + 3z + 3/2 = 5z + 2
=> (1/2 + 3)z + 2 + 3/2 = 5z + 2
=> 7/2z + 7/2 = 5z + 2
=> 7/2 - 2 = 5z - 7/2z
=> 1.5 = 1.5z
=> z = 1

So,
LM = 1/2 + 2 = 2.5
MN = 3 + 1.5 = 4.5
LN = 5 + 2 = 7

You can check now LM + MN = 2.5 + 4.5 = 7 (which is LN)