Question

1. 5x, 10y - 40, 2y (diagram of intersecting lines) 8. 4y, x, 8y - 116 (diagram of intersecting lines) (partial: x+30, 15x, +30)

Explanation:

Problem 6 (assuming we solve for \( y \) first, then maybe \( x \); let's focus on the parallel lines and alternate exterior or corresponding angles)

Step1: Identify equal angles (corresponding or alternate)

From the diagram, \( 10y - 40 \) and \( 2y \) seem to be equal (since the lines are parallel, alternate or corresponding angles). So set up the equation:
\( 10y - 40 = 2y \)

Step2: Solve for \( y \)

Subtract \( 2y \) from both sides:
\( 10y - 2y - 40 = 0 \)
\( 8y - 40 = 0 \)

Add 40 to both sides:
\( 8y = 40 \)

Divide by 8:
\( y = \frac{40}{8} = 5 \)

Step3: Now find \( x \) (assuming \( 5x \) and \( 10y - 40 \) are supplementary or equal? Wait, maybe \( 5x \) and \( 2y \) are equal? Wait, if \( y = 5 \), \( 2y = 10 \), so \( 5x = 10 \)? Wait, no, maybe \( 5x \) and \( 10y - 40 \) are equal. Let's check: \( 10y - 40 = 10(5) - 40 = 50 - 40 = 10 \), so \( 5x = 10 \), so \( x = 2 \). But maybe the problem is just to solve for \( y \) first. Let's confirm the angle relationship. Since the two slanted lines are parallel (same direction), the angles \( 10y - 40 \) and \( 2y \) are alternate exterior angles, so equal. So \( y = 5 \), and if we need \( x \), \( 5x = 10y - 40 = 10 \), so \( x = 2 \).

Problem 8 (parallel lines, so \( 4y \) and \( 8y - 116 \) are equal? Wait, no, \( 4y \) and \( 8y - 116 \): since the lines are parallel, maybe alternate interior angles? Wait, \( 4y \) and \( 8y - 116 \) should be equal (if the transversal cuts parallel lines, alternate interior angles). So:

Step1: Set up the equation for \( y \)

\( 4y = 8y - 116 \)

Step2: Solve for \( y \)

Subtract \( 4y \) from both sides:
\( 0 = 4y - 116 \)

Add 116 to both sides:
\( 4y = 116 \)

Divide by 4:
\( y = \frac{116}{4} = 29 \)

Step3: Find \( x \) (since \( 4y \) and \( x \) are supplementary? Wait, \( 4y + x = 180^\circ \)? Wait, no, \( 4y \) and \( x \) are adjacent angles on a straight line, so they are supplementary. So \( x = 180 - 4y \). Substitute \( y = 29 \):

\( x = 180 - 4(29) = 180 - 116 = 64 \)

Answer:

s:

• Problem 6: \( y = 5 \), \( x = 2 \) (if needed)
• Problem 8: \( y = 29 \), \( x = 64 \)

(Note: The exact solution depends on the angle relationships, but based on the diagram, the above steps assume parallel lines and corresponding/alternate angles or linear pairs.)