If given a diagram of two pairs of vertical angles, where one vertical angle on the bottom is narrow and can be defined as 3x, and the wider angle on the right is 5x - 13, and the wider angle on the left is 8y + 6, you can solve for x and y in the following manner:
We can prove that the narrow angle on top is 3x, since two vertical angles always equal each other. We also know that the two angles on either side of the diagram are linear pairs (draw a diagram if you wish). This means that they add up to 180 degrees. We can form this equation: 3x + 5x - 13 = 180.
Combine like terms.
8x - 13 = 180
Add 13 to both sides.
8x = 193
Divide both sides by 8.
x = 24.125 (or 24 and 1/8)
Now we just need to solve for y. We can do this with the following equation:
8y + 6 + 3x = 180
We can form this equation because we know the two linear pairs add to 180 degrees. Now we need to eliminate a variable. Plug the previously solved x value into the equation.
8y + 6 + 3(24.125) = 180
8y + 6 + 72.375 = 180
Combine like terms.
8y + 78.375 = 180
Subtract 78.375 from both sides.
8y = 101.625
Divide both sides by 8.
y = 12.703125