Question 183149
Let the present age of the man = {{{m}}}
Let the present age of the boy = {{{b}}}
Let the present age of the girl = {{{g}}}
2 years ago:
The man's age was {{{m - 2}}}
The boys aghe was {{{b - 2}}}
The girls age was {{{g - 2}}}
In 8 years:
The man's age will be {{{m + 8}}}
The boy's age will be {{{b + 8}}}
The girl's age will be {{{g + 8}}}
given:
(1) {{{b = g + 6}}}
(2) {{{m - 2 = 3*(b - 2 + g - 2)}}}
(3) {{{m + 8 = b + 8 + g + 8 + 10}}}
-------------------------
(1) {{{b = g + 6}}}
(2) {{{m - 2 = 3b - 6 + 3g - 6}}}
(2) {{{m = 3b + 3g - 10}}}
--------------------------
(3) {{{m + 8 = b + 8 + g + 8 + 10}}}
(3) {{{m = b + g + 18}}}
--------------------------
Multiply both sides of (3) by {{{3}}} and subtract (2) from (3)
(3) {{{3m = 3b + 3g + 54}}}
(2) {{{-m = -3b - 3g + 10}}}
(4) {{{2m = 64}}}
(4) {{{m = 32}}}
Substitute this result back into (2)
(2) {{{m = 3b + 3g - 10}}}
(2) {{{32 = 3b + 3g - 10}}}
(2) {{{3b + 3g = 42}}}
(2) {{{b + g = 14}}}
and, from (1),
(1) {{{b = g + 6}}}
(1) {{{b - g = 6}}}
Now add (1) and (2)
(2) {{{b + g = 14}}}
(1) {{{b - g = 6}}}
(5) {{{2b = 20}}}
(5) {{{b = 10}}}
Putting this back in (1),
(1) {{{10 = g + 6}}}
(1) {{{g = 4}}}
The man is 32, the boy is 10, and the girl is 4
check answers:
(1) {{{b = g + 6}}}
(2) {{{m = 3b + 3g - 10}}}
(3) {{{m = b + g + 18}}}
--------------------------
(1) {{{b = g + 6}}}
(1) {{{10 = 4 + 6}}}
(1) {{{10 = 10}}}
(2) {{{m = 3b + 3g - 10}}}
(2) {{{32 = 3*10 + 3*4  - 10}}}
(2) {{{32 = 30 + 12 -10}}}
(2) {{{32 = 32}}}
(3) {{{m = b + g + 18}}}
(3) {{{32 = 10 + 4 + 18}}}
(3) {{{32 = 32}}}
OK