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.

Answer 1

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.