Question

7 use the diagram to answer the question below.
xm = 10x + 6 and my = 6x + 22
if m is the midpoint of segment \\(overline{xy}\\), solve for \\(overline{xy}\\).
\\(xy = 46\\)

Explanation:

Step1: Set XM equal to MY (midpoint property)

Since \( M \) is the midpoint, \( XM = MY \). So we set up the equation:
\( 10x + 6 = 6x + 22 \)

Step2: Solve for x

Subtract \( 6x \) from both sides:
\( 10x - 6x + 6 = 22 \)
\( 4x + 6 = 22 \)

Subtract 6 from both sides:
\( 4x = 22 - 6 \)
\( 4x = 16 \)

Divide both sides by 4:
\( x = \frac{16}{4} = 4 \)

Step3: Find XM (or MY)

Substitute \( x = 4 \) into \( XM = 10x + 6 \):
\( XM = 10(4) + 6 = 40 + 6 = 46 \)

Step4: Find XY (midpoint implies XY = 2*XM)

Since \( M \) is the midpoint, \( XY = XM + MY = 2 \times XM \) (because \( XM = MY \)).
So \( XY = 2 \times 46 = 92 \)? Wait, wait, no—wait, let's check again. Wait, if \( x = 4 \), then \( XM = 10(4)+6 = 46 \), \( MY = 6(4)+22 = 24 + 22 = 46 \). Then \( XY = XM + MY = 46 + 46 = 92 \). Wait, the initial box had 46, but that's wrong. Let's redo:

Wait, step1: Midpoint means \( XM = MY \). So \( 10x + 6 = 6x + 22 \).
Step2: \( 10x - 6x = 22 - 6 \) → \( 4x = 16 \) → \( x = 4 \).
Step3: \( XM = 10(4) + 6 = 46 \), \( MY = 6(4) + 22 = 46 \).
Step4: \( XY = XM + MY = 46 + 46 = 92 \). So the initial box was incorrect. Let's correct:

(Correct):

Step1: Midpoint implies \( XM = MY \)

\( 10x + 6 = 6x + 22 \)

Step2: Solve for \( x \)

\( 10x - 6x = 22 - 6 \)
\( 4x = 16 \)
\( x = 4 \)

Step3: Calculate \( XM \) (or \( MY \))

\( XM = 10(4) + 6 = 46 \) (same as \( MY = 6(4) + 22 = 46 \))

Step4: Calculate \( XY \)

\( XY = XM + MY = 46 + 46 = 92 \)

Answer:

\( \boxed{92} \)