ABCD is an isosceles trapezoid with legs AB and CD and base BC, If the length of overline AB = 10y - 16 , the length of overline BC = 4y - 6 and the length of overline CD = 8v - 4 . find the value of y. Make sure to show ALL of your work in order to receive full credit.
Since ABCD is an isosceles trapezoid, its legs AB and CD have the same length. Therefore, 10y - 16 = 8v - 4.
We are given that AB = 10y - 16 and BC = 4y - 6. If we subtract BC from AB, we should get the length of AD. So, AD = (10y - 16) - (4y - 6) = 10y - 4y - 16 + 6 = 6y - 10.
We are also given that CD = 8v - 4. Since AD = CD, we can say 6y - 10 = 8v - 4.
Now we have two equations: 10y - 16 = 8v - 4 and 6y - 10 = 8v - 4.
Rearranging the second equation, we get 6y - 8v = 6.
Now we have a system of two equations:
10y - 16 = 8v - 4
6y - 8v = 6
We can solve this system using any method, but let's use substitution. Solving the second equation for 6y, we get 6y = 8v + 6.
Substituting 8v + 6 for 6y in the first equation, we get 10(8v + 6) - 16 = 8v - 4.
Distributing, we get 80v + 60 - 16 = 8v - 4.
Combining like terms, we get 80v + 44 = 8v - 4.
Subtracting 8v from both sides, we get 72v + 44 = -4.
Subtracting 44 from both sides, we get 72v = -48.
Dividing both sides by 72, we get v = -48/72 = -2/3.
Now that we have the value of v, we can substitute it back into one of the original equations to find the value of y. Let's use 6y - 8v = 6.
Substituting -2/3 for v, we get 6y - 8(-2/3) = 6.
Simplifying, we get 6y + 16/3 = 6.
Subtracting 16/3 from both sides, we get 6y = 6 - 16/3.
Combining like terms and multiplying both sides by 3, we get 18y = 18 - 16.
Simplifying, we get 18y = 2.
Dividing both sides by 18, we get y = 2/18 = 1/9.
Therefore, the value of y is 1/9.