The sum of three integers is 270. the sum of the first and second integers exceeds the third by 98. the third integer is 47 less than the first. find the three integers.

I'm going to do this problem with mathematical equations (I chose x to represent the first, y as the second, and z as the third)
so according to the first sentence, x+y+z=270
the second sentence is saying x+y=z+98
and the third sentence says z-47=x

I would first plug x in to my middle equation
(z-47)+y=z+98
then add 47 to both sides to get z+y=z+145
In this case we are lucky because as we go to subtract z they cancel out and we get y=145 

Now plug y and x (in terms of z) into the first equation 
(z-47)+145+z=270
subtract 47 from 145 to get 98
 subtract that from 270 to give you z+z=172 
simplify 2z=172
divide by 2 to get z=86

Now go back and plug in z to the bottom equation,
86-47=x so x is equal to 39

Overall we got x=39, y=145 and z=86
To make sure these answers are correct we want to plug the back into our original equations.
39+145+86=270 True!
39+145=184=86+98 True!
and finally 86-47=39 True!
So we know that these numbers are correct

*Just so you know it doesn't really matter which equation you plug in you substituted values into, or which one you solve for first, it's all just a matter of preference