Hello Sydney. Let's have Adam's present age be represented by A and Bob's present age be represented by B. The first scenario takes place ten years from the present; as such we can say Adam's age is A+10 and Bob's age is also B+10. At the same time, Adam was twice as old as Bob, which means however old Adam was, Bob's age was that if it was multiplied by two. So the first equation can be written as...
A+10 = 2*(B+10).....................................(1)
The second scenario takes place five years ago; as such we can say Adam's age is A-5 and Bob's age is also B-5. At the same time, Adam was three times as old as Bob, which means however old Adam was, Bob's age was that if it was multiplied by three. So the second equation can be written as...
A-5 = 3*(B-5)...........................................(2)
If we simplify both equations, we get...
A+10 = 2B+20..........................................(1)
A-5 = 3B-15..............................................(2)
And if we rearrange both of these equations to be in standard form we get...
A-2B = 10.................................................(1)
A-3B = -10................................................(2)
At this point you can use either substitution or elimination to solve this system of equations, but I'll use substitution since this problem is easier to solve it this way. So now let's solve Eq. 1 for A and plug it in to Eq. 2 as such...
A = 2B+10.................................................(1)
plug it in to Eq. 2 and solve ---------> (2B+10) -3B = -10.........................................(2)
-B + 10 = -10
-B = -20
B = 20 <-----------------Bob's Age
So now that we have Bob's age, we can plug it in into either Eq. 1 or 2 and get Adam's age. I'll substitute it into Eq. 2 and get...
A - 3(20) = -10
A - 60 = -10
A = 50 <----------------- Adam's Age
So by solving this system of equations we get that right now, Adam is 50 years old and Bob is 20 years old.
