Question

point m lies between points l and n on (overline{ln}). if (ln = 12x + 16), what is the length of (overline{ln}) in units? (overline{ln}) is a line segment with (l) on the left, (m) in the middle, (n) on the right. the length from (l) to (m) is labeled (10x + 8), and from (m) to (n) is labeled (5x - 4). the options are: 16 units, 40 units, 48 units, 64 units.

Explanation:

Step1: Set up the equation for LN

Since M lies between L and N, \( LN = LM + MN \). So, \( 12x + 16=(10x + 8)+(5x - 4) \).

Step2: Simplify the right - hand side

Simplify \( (10x + 8)+(5x - 4) \): \( 10x+5x + 8 - 4=15x + 4 \). Now our equation is \( 12x + 16 = 15x+4 \).

Step3: Solve for x

Subtract \( 12x \) from both sides: \( 16=15x - 12x+4 \), which simplifies to \( 16 = 3x+4 \). Then subtract 4 from both sides: \( 16 - 4=3x \), so \( 12 = 3x \). Divide both sides by 3: \( x = 4 \).

Step4: Find the length of MN

We know that \( MN = 5x - 4 \). Substitute \( x = 4 \) into the expression for MN: \( MN=5\times4 - 4=20 - 4 = 16 \)? Wait, no, wait. Wait, the question is about LN? Wait, no, the options are 16,40,48,64. Wait, let's re - check. Wait, \( LN=12x + 16 \). Substitute \( x = 4 \) into \( LN \): \( 12\times4+16=48 + 16=64 \)? Wait, no, wait, let's check the equation again.

Wait, \( LM = 10x + 8 \), \( MN = 5x - 4 \), and \( LN=LM + MN \). So \( 12x + 16=(10x + 8)+(5x - 4) \). Let's re - solve the equation:

\( 12x+16 = 10x + 8+5x - 4\)

\( 12x+16=15x + 4\)

Subtract \( 12x \) from both sides: \( 16 = 3x+4\)

Subtract 4: \( 12 = 3x\), so \( x = 4 \).

Now, \( LN=12x + 16=12\times4+16 = 48 + 16=64 \)? Wait, but let's check \( LM \) and \( MN \) with \( x = 4 \):

\( LM=10\times4 + 8=40 + 8 = 48\)

\( MN=5\times4-4 = 20 - 4=16\)

And \( 48 + 16=64\), which matches \( LN = 12x + 16=64 \) when \( x = 4 \). But wait, the question is "what is the length of \( \overline{LN} \) in units?". So the length of LN is 64 units? Wait, but let's check the options. The options are 16,40,48,64. So 64 is an option.

Wait, maybe I made a mistake in the equation. Wait, is \( LN=LM + MN \)? Yes, because M is between L and N. So \( LM+MN = LN \). So \( 10x + 8+5x - 4=12x + 16 \). Combining like terms: \( 15x + 4=12x + 16 \). Subtract \( 12x \): \( 3x+4 = 16 \). Subtract 4: \( 3x = 12 \), so \( x = 4 \). Then \( LN=12\times4+16 = 48 + 16=64 \). So the length of LN is 64 units.

Answer:

64 units