Question

m is the midpoint of ( overline{ln} ). if ( lm = 6x + 10 ) and ( ln = 16x + 3 ), what is lm?
simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use the midpoint property

Since \(M\) is the midpoint of \(\overline{LN}\), then \(LN = 2LM\).
Given \(LM=6x + 10\) and \(LN = 16x+3\), we substitute into the equation \(LN = 2LM\):
\(16x+3=2(6x + 10)\)

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(16x+3 = 12x+20\)
Subtract \(12x\) from both sides: \(16x-12x+3=12x - 12x+20\), so \(4x+3=20\)
Subtract 3 from both sides: \(4x+3 - 3=20 - 3\), then \(4x=17\)
Divide both sides by 4: \(x=\frac{17}{4}\)

Step3: Find the value of \(LM\)

Substitute \(x = \frac{17}{4}\) into \(LM=6x + 10\)
\(LM=6\times\frac{17}{4}+10=\frac{102}{4}+10=\frac{51}{2}+10=\frac{51 + 20}{2}=\frac{71}{2}=35\frac{1}{2}\)

Answer:

\(35\frac{1}{2}\)