Question

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

Explanation:

Step1: Use midpoint property

Since \( L \) is the midpoint of \( \overline{KM} \), \( KM = 2 \times LM \). So we have the equation \( 6x - 3 = 2(x + 6) \).

Step2: Solve for \( x \)

Expand the right side: \( 6x - 3 = 2x + 12 \).
Subtract \( 2x \) from both sides: \( 6x - 2x - 3 = 12 \), which simplifies to \( 4x - 3 = 12 \).
Add 3 to both sides: \( 4x = 12 + 3 = 15 \).
Divide both sides by 4: \( x=\frac{15}{4} \).

Step3: Find \( KM \)

Substitute \( x = \frac{15}{4} \) into \( KM = 6x - 3 \).
\( KM = 6\times\frac{15}{4}-3=\frac{90}{4}-3=\frac{45}{2}-3=\frac{45 - 6}{2}=\frac{39}{2}=19.5 \) (or as a fraction \( \frac{39}{2} \) or mixed number \( 19\frac{1}{2} \)). Wait, let's check the calculation again. Wait, \( 6x-3 \) when \( x = 15/4 \): \( 6(15/4)=90/4 = 45/2 = 22.5 \), then \( 22.5 - 3 = 19.5 \)? Wait, no, 6(15/4) is (6*15)/4 = 90/4 = 45/2 = 22.5, then 22.5 - 3 = 19.5? Wait, but let's re - solve the equation \( 6x - 3=2(x + 6) \).

Expanding: \( 6x-3 = 2x + 12 \)

Subtract \( 2x \): \( 4x-3 = 12 \)

Add 3: \( 4x=15 \), so \( x = 15/4 \)

Then \( KM=6x - 3=6*(15/4)-3=(90/4)-3=(45/2)-3=(45 - 6)/2 = 39/2=19.5 \) or \( 19\frac{1}{2} \) or \( \frac{39}{2} \). Wait, but let's check the midpoint property again. If \( L \) is the midpoint, then \( KL = LM \) and \( KM=KL + LM = 2LM \). So \( KM = 2LM \), so \( 6x - 3=2(x + 6) \). Let's solve the equation correctly:

\( 6x-3 = 2x + 12 \)

\( 6x-2x=12 + 3 \)

\( 4x=15 \)

\( x = 15/4 \)

Then \( LM=x + 6=15/4+6=15/4 + 24/4=39/4 \)

Then \( KM = 2LM=2*(39/4)=39/2 = 19.5 \). Yes, that's correct.

Answer:

\( \frac{39}{2} \) (or \( 19\frac{1}{2} \) or \( 19.5 \))