Question

what is the length of $overline{ln}$, when m is the midpoint of $overline{ln}$? diagram: points l, m, n on a line; $overline{lm}$ labeled $5x - 7$, $overline{mn}$ labeled $3(x + 7)$; multiple - choice options: 14, 28, 63, 126 (28 is selected)

Explanation:

Step1: Set LM = MN (midpoint definition)

Since M is the midpoint of \(\overline{LN}\), \(LM = MN\). So we set up the equation: \(5x - 7 = 3(x + 7)\)

Step2: Solve for x

Expand the right side: \(5x - 7 = 3x + 21\)
Subtract \(3x\) from both sides: \(5x - 3x - 7 = 21\) → \(2x - 7 = 21\)
Add 7 to both sides: \(2x = 21 + 7\) → \(2x = 28\)
Divide by 2: \(x = \frac{28}{2} = 14\)

Step3: Find length of LM

Substitute \(x = 14\) into \(LM = 5x - 7\): \(5(14) - 7 = 70 - 7 = 63\)

Step4: Find length of LN

Since M is the midpoint, \(LN = 2 \times LM\). So \(LN = 2 \times 63 = 126\)

Answer:

126