let y = the age of the kids today.
your equations are:
x = 6y
x-2 = 10 * (y-2)
x+6 = 3 * (y+6)
in order to solve this, all equations must have a common solution.
unfortunately they don't.
if you solve for a common solution for the first 2 equations, you will get:
x = 27 and y = 4.5
if you solve for a common solution for the first and third equation, you will get:
x = 24 and y = 4
if you solve for a common solution for the last 2 equations, you will get:
x = 24.857 and y = 4.286
this means that you have no common solution for all 3 equations, therefore you have no solution.
this can be seen graphically below:
x-6y is the red equation and red line on the graph.
x-2 = 10 * (6-2) is the blue equation and blue line on the graph.
x+6 = 3 * (6+6) is the green equation and green line on the graph.
there is a common solution between red and blue.
there is a common solution between red and green.
therr is a common solution between blue and green.
but, there is not a common solution between all 3.
a common solution to all 3 equations cannot be found, therefore there is no solution.
one of those equations is not being presented correctly.
for example, assume 24 and 4 needed to be the answer.
first equation is ok because 24 = 6 * 4.
third equation is ok because 30 = 3 * 10
second equation is not ok.
in order for second equation to be ok, let x = 24 and let y = 4 and evaluate the equation as is:
you will get 24 - 2 = 10 * (y-2).
this becomes 22 = 10 * (2)
you will get 22 = 20 which is not correct.
look at the ages today.
x = 24 and y = 4
2 years ago, x = 22 and y = 2
this makes x equal to 11 * 2.
the equaton should be:
x-2 = 11 * (y-2)
now, when you replace x with 24 and y with 4, you get:
22 = 11 * 2 which becomes 22 = 22 which is correct.
so, one of the equations is not in synch with the other equations.
not sure which one, but one of them is definitely off.
check your inputs to see that you got them correctly.
if so, then there is no solution common to all 3 equations.
another problem is interpretation.
if the sum of the parents age is 24, then they must have been very young.
too young, in fact, to have children.
you could have been talking about their average age.
in this case, everything would be doubled and you would wind up with the same problem if you assume 2 parents and 2 children.
all in all, you have a pretty messed up problem here.
there is no common solution as it is presented.
it does not make sense unless there is more information available that is not presented.