Question

solve for x:
diagram showing two intersecting lines with angles labeled ( 8x + 36^circ ) (blue) and ( 5x + 60^circ ) (green), forming a linear pair
( x = ) empty box

Explanation:

Step1: Identify supplementary angles

The two angles \(8x + 36^\circ\) and \(5x + 60^\circ\) are supplementary (they form a linear pair), so their sum is \(180^\circ\).
\[
(8x + 36) + (5x + 60) = 180
\]

Step2: Combine like terms

Combine the \(x\)-terms and the constant terms:
\[
13x + 96 = 180
\]

Step3: Solve for \(x\)

Subtract 96 from both sides:
\[
13x = 180 - 96
\]
\[
13x = 84
\]
Wait, no, wait. Wait, 36 + 60 is 96? Wait, 36 + 60 is 96? Wait, 180 - 96 is 84? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, 8x + 36 + 5x + 60 = 180. So 13x + 96 = 180. Then 13x = 180 - 96 = 84? Wait, 180 - 96 is 84? Then x = 84 / 13? No, that's not right. Wait, maybe the angles are vertical or something else? Wait, no, looking at the diagram, the two angles are adjacent and form a linear pair? Wait, no, maybe they are alternate interior angles? Wait, no, the diagram shows two lines intersecting, so maybe the two angles are supplementary? Wait, no, maybe I misread the angles. Wait, the blue angle is \(8x + 36\) and the green angle is \(5x + 60\). Wait, maybe they are supplementary? Wait, 8x + 36 + 5x + 60 = 180. So 13x + 96 = 180. 13x = 84. x = 84 / 13 ≈ 6.46? That doesn't make sense. Wait, maybe the angles are equal? Wait, no, maybe they are vertical angles? Wait, no, vertical angles are equal. Wait, maybe the two angles are supplementary? Wait, no, maybe I made a mistake. Wait, let's check again. Wait, 8x + 36 and 5x + 60. If they are supplementary, then 8x + 36 + 5x + 60 = 180. 13x + 96 = 180. 13x = 84. x = 84 / 13 ≈ 6.46. But that's not an integer. Wait, maybe the angles are equal? Wait, no, maybe the diagram is different. Wait, maybe the two angles are adjacent and form a linear pair, but maybe I misread the expressions. Wait, maybe the blue angle is \(8x + 36\) and the green angle is \(5x + 60\), and they are supplementary. Wait, 8x + 36 + 5x + 60 = 180. 13x = 84. x = 84 / 13. No, that's not right. Wait, maybe the angles are vertical angles? Wait, no, vertical angles are equal. So 8x + 36 = 5x + 60. Then 3x = 24. x = 8. Ah! That's it. Maybe the two angles are vertical angles. So 8x + 36 = 5x + 60. Then 8x - 5x = 60 - 36. 3x = 24. x = 8. Yes, that makes sense. So I made a mistake earlier, thinking they were supplementary, but actually, they are vertical angles, so they are equal. So:

Step1: Set angles equal (vertical angles)

\[
8x + 36 = 5x + 60
\]

Step2: Subtract \(5x\) from both sides

\[
3x + 36 = 60
\]

Step3: Subtract 36 from both sides

\[
3x = 24
\]

Step4: Divide by 3

\[
x = 8
\]

Answer:

8