Answer:
The largest even integer is <u>14</u>.
Step-by-step explanation:
Given:
Three times the second of three consecutive even integers is twelve less than twice the sum of the first and third integers.
Now, to find the largest even integers.
As, 2 is an even number.
Let an even integer be
So, the three consecutive even integers are:
![2x,\ 2x+2,\ and\ 2x+4.](https://tex.z-dn.net/?f=2x%2C%5C%202x%2B2%2C%5C%20and%5C%202x%2B4.)
Now, we have to add two to get next even number.
Thus, 3 times the second one of these is:
![3(2x + 2) \\= 6x + 6](https://tex.z-dn.net/?f=3%282x%20%2B%202%29%20%5C%5C%3D%206x%20%2B%206)
Now, the sum of the first and third ones are:
![2x + 2x + 4 \\= 4x + 4](https://tex.z-dn.net/?f=2x%20%2B%202x%20%2B%204%20%5C%5C%3D%204x%20%2B%204)
So, the twice the sum of the first and third integers is less than twelve:
![2(4x+4)-12\\=8x+8-12\\=8x-4](https://tex.z-dn.net/?f=2%284x%2B4%29-12%5C%5C%3D8x%2B8-12%5C%5C%3D8x-4)
Now, 3 times the second is 12 less than twice the sum of the other two:
![6x+6=8x-4](https://tex.z-dn.net/?f=6x%2B6%3D8x-4)
<em>Getting the variables on one side and the number on other:</em>
![6+4=8x-6x](https://tex.z-dn.net/?f=6%2B4%3D8x-6x)
![10=2x](https://tex.z-dn.net/?f=10%3D2x)
<em>Dividing both sides by 2 we get:</em>
![5=x](https://tex.z-dn.net/?f=5%3Dx)
![x=5.](https://tex.z-dn.net/?f=x%3D5.)
Now, the even numbers we get are:
![2x=2\times 5=10\\2x+2=2\times 5+2=10+2=12\\2x+4=2\times 5+4=10+4=14.](https://tex.z-dn.net/?f=2x%3D2%5Ctimes%205%3D10%5C%5C2x%2B2%3D2%5Ctimes%205%2B2%3D10%2B2%3D12%5C%5C2x%2B4%3D2%5Ctimes%205%2B4%3D10%2B4%3D14.)
<em>Thus, the third number is the largest number which is 14.</em>
Therefore, the largest even integer is 14.