what does the opposite mean?

Question

 find four consecutive even integers such as that -4 times the sum of first and the fourth is 6 greater than the opposite of the sum of the second and the third

Answer 1

It's not a technical mathematical term, but in this case it probably means "negative".

 

So you'd have 4 consecutive even numbers:  n, n+2, n+4, n+6.  The statement can be written as follows:

 

-4(n + n+6) = 6 + -(n+2 + n+4)

 

Let's solve that:

 

-4(2n + 6) = 6 - (2n + 6)

-8n - 24 = 6 - 2n - 6

-8n - 24 = -2n

-6n = 24

n = -4

 

So the integers would be -4, -2, 0, and 2

 

Let's see if it works: 

 

"-4 times the sum of first and the fourth" = -4 (-4 + 2) = -4 (-2) = 8

"the opposite of the sum of the second and the third" = - (-2 + 0) = 2

 

8 is 6 more than 2, so that's correct.

 

 

 

 

Answer 2

Opposite means the same as "the negative."

 

The four consecutive even integers

n, n + 2, n + 4, n + 6

 

-4 times the sum of the first and fourth

-4(n + n + 6)

 

6 greater than the opposite of the sum of the second and third

-(n + 2 + n + 4) + 6

 

-4(n + n + 6) = -(n + 2 + n + 4) + 6

-4(2n + 6) = -(2n + 6) + 6

-8n - 24 = -2n - 6 + 6

-8n - 24 = -2n

-24 = 6n

-4 = n

 

What are the other three integers?