the sum of an even integer and twice the next greater even integer is the same as the difference between five times the greater even integer is 14. Find the integers. There are two solutions
I can't find the integers until you clarify what
the difference between five times the greater even integer is 14
means
To answer Steve's earlier statement, "the difference between five times the greater even integer is 14" means "5(x+2) =14"
To solve this problem, we need to break it down into steps and use algebraic expressions to represent the different parts of the problem.
Let's start by representing the two even integers. We can call the first even integer "x" and the next greater even integer "x + 2".
According to the problem, the sum of an even integer and twice the next greater even integer is the same as the difference between five times the greater even integer and 14.
This can be translated into the following equation:
x + 2(x + 2) = 5(x + 2) - 14
Now, we can simplify and solve the equation:
x + 2x + 4 = 5x + 10 - 14
3x + 4 = 5x - 4
4 + 4 = 5x - 3x
8 = 2x
x = 4
So, the first even integer, x, is 4.
Now let's find the next greater even integer by adding 2 to x:
x + 2 = 4 + 2 = 6
Therefore, the first solution is x = 4 and the second greater even integer is 6.
Checking the answer:
The sum of the even integer (4) and twice the next greater even integer (2 * 6) is:
4 + 2 * 6 = 4 + 12 = 16
The difference between five times the greater even integer (5 * 6) and 14 is:
5 * 6 - 14 = 30 - 14 = 16
Both results are equal to 16, confirming that the integers 4 and 6 are a valid solution.
Since there are two solutions, we can also find the second solution by increasing the first even integer, x, by another 2:
x + 2 + 2 = 4 + 2 + 2 = 8
Therefore, the second solution is x = 4 and the second greater even integer is 8.
Checking the answer:
The sum of the even integer (4) and twice the next greater even integer (2 * 8) is:
4 + 2 * 8 = 4 + 16 = 20
The difference between five times the greater even integer (5 * 8) and 14 is:
5 * 8 - 14 = 40 - 14 = 26
Again, both results are equal to 26, confirming that the integers 4 and 8 are the second valid solution.
So, the two possible sets of integers that satisfy the given conditions are (4, 6) and (4, 8).