Question

if the first and third of three consecutive odd integers are added, the result is 75 less than five times the second integers. find the third integer. please show work

Answer 1

The third integer is 27

Explanation:

Given

Represent the consecutive odd numbers with
`T_1, T_2; and\ T_3`

Since they are consecutive odds, then

`T_2 = 2 + T_1`

`T_3 = 2 + T_2`

From the first statement in the question, we have that

`T_1 + T_3 = 5 T_2 - 75`

Required

Find the third integer

Recall that

Substitute
`T_2 = 2 + T_1` in
`T_3 = 2 + T_2`

`T_3 = 2 + 2 + T_1`

`T_3 = 4 + T_1`

Substitute
`T_3 = 4 + T_1` and
`T_2 = 2 + T_1` in
`T_1 + T_3 = 5 T_2 - 75`

`T_1 + 4 + T_1 = 5(2 + T_1) - 75`

Open Brackets

`T_1 + 4 + T_1 = 10 + 5T_1 - 75`

Collect like terms

`T_1 + T_1 + 4 = 5T_1 - 75 + 10`

`2T_1 + 4 = 5T_1 - 65`

Collect like terms

`2T_1 - 5T_1 = - 65 - 4`

`-3T_1 = - 69`

Divide both sides by -3

`(-3T_1)/(-3) = (- 69)/(-3)`

`T_1 = 23`

Recall that
`T_3 = 4 + T_1`

`T_3 = 4 + 23`

`T_3 = 27`

The third integer is 27