Question

1. m is the midpoint of (overline{ab}). am = 4x - 13, mb = 2x + 3. solve for x and find ab.

x = ____
ab = ____
(diagram of line segment with points a, m, b)

Explanation:

Step1: Set AM equal to MB (midpoint property)

Since \( M \) is the midpoint of \( \overline{AB} \), \( AM = MB \). So we set up the equation:
\( 4x - 13 = 2x + 3 \)

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides:
\( 4x - 2x - 13 = 3 \)
\( 2x - 13 = 3 \)

Add 13 to both sides:
\( 2x = 3 + 13 \)
\( 2x = 16 \)

Divide both sides by 2:
\( x = \frac{16}{2} \)
\( x = 8 \)

Step3: Find \( AM \) and \( MB \)

Substitute \( x = 8 \) into \( AM = 4x - 13 \):
\( AM = 4(8) - 13 = 32 - 13 = 19 \)

Substitute \( x = 8 \) into \( MB = 2x + 3 \):
\( MB = 2(8) + 3 = 16 + 3 = 19 \)

Step4: Find \( AB \)

Since \( AB = AM + MB \), and \( AM = MB = 19 \):
\( AB = 19 + 19 = 38 \)

Answer:

\( x = 8 \)
\( AB = 38 \)