Final answer:
To solve this problem, we need to set up an equation and solve for the values of the consecutive even integers.
Step-by-step explanation:
To solve this problem, let's assume that the first even integer is x. Since the numbers are consecutive and even, the second even integer would be x + 2, and the third even integer would be x + 4.
The problem states that the sum of triple the first number, double the second, and four times the third is 38. Translating this information into an equation, we have:
3x + 2(x+2) + 4(x+4) = 38
Simplifying the equation, we get:
3x + 2x + 4x + 2(2) + 4(4) = 38
This can be further simplified to:
9x + 4 + 16 = 38
9x + 20 = 38
Subtracting 20 from both sides gives us:
9x = 18
Dividing both sides by 9, we find:
x = 2
Now we can find the other two consecutive even integers. The second integer is x + 2, which is 2 + 2 = 4. The third integer is x + 4, which is 2 + 4 = 6.
Therefore, the three consecutive even integers are 2, 4, and 6.