If the first and third of three consecutive even integers are added, the result is 12 less than three times the second integer. Find the integers.
To solve this problem, let's break down the given information step by step.
Let's assume the first even integer as n.
According to the given information, the second even integer would be consecutive to the first even integer. So, the second even integer can be represented as (n + 2).
Similarly, the third even integer would also be consecutive to the second even integer. Therefore, the third even integer can be represented as (n + 4).
Now, let's translate the given information into an equation:
"If the first and third of three consecutive even integers are added, the result is 12 less than three times the second integer."
In equation form, this can be written as:
n + (n + 4) = 3(n + 2) - 12.
Now, let's solve the equation step by step:
n + n + 4 = 3n + 6 - 12
Combining like terms:
2n + 4 = 3n - 6
Bringing all the terms with n to the left side and the constant terms to the right side:
2n - 3n = -6 - 4
Simplifying:
-n = -10
Multiplying both sides of the equation by -1 to isolate n:
n = 10
Now, we have the value of the first even integer, which is n = 10.
To find the second and third even integers, we substitute the value of n back into our expressions:
Second even integer = n + 2 = 10 + 2 = 12
Third even integer = n + 4 = 10 + 4 = 14
So, the three consecutive even integers are 10, 12, and 14.