Answer:
The largest even integer is 14.
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 [tex]2x.[/tex]
So, the three consecutive even integers are:
[tex]2x,\ 2x+2,\ and\ 2x+4.[/tex]
Now, we have to add two to get next even number.
Thus, 3 times the second one of these is:
[tex]3(2x + 2) \\= 6x + 6[/tex]
Now, the sum of the first and third ones are:
[tex]2x + 2x + 4 \\= 4x + 4[/tex]
So, the twice the sum of the first and third integers is less than twelve:
[tex]2(4x+4)-12\\=8x+8-12\\=8x-4[/tex]
Now, 3 times the second is 12 less than twice the sum of the other two:
[tex]6x+6=8x-4[/tex]
Getting the variables on one side and the number on other:
[tex]6+4=8x-6x[/tex]
[tex]10=2x[/tex]
Dividing both sides by 2 we get:
[tex]5=x[/tex]
[tex]x=5.[/tex]
Now, the even numbers we get are:
[tex]2x=2\times 5=10\\2x+2=2\times 5+2=10+2=12\\2x+4=2\times 5+4=10+4=14.[/tex]
Thus, the third number is the largest number which is 14.
Therefore, the largest even integer is 14.
