Three times Diana's age is $17$ years more than twice Jim's age. The sum of their ages is $13$ years less than their fathers age, which is three times Jims age. What are the children's ages?

Let Diana's age be D and Jim's age be J. The problem yields three equations: 3D equals 2J plus 17, Father's age equals 3J, and D plus J equals 3J minus 13. Simplifying the third equation gives D equals 2J minus 13. Substituting this into the first equation gives 3 times (2J minus 13) equals 2J plus 17, which simplifies to 6J minus 39 equals 2J plus 17, yielding J equals 14. Plugging J back into D equals 2J minus 13 gives Diana's age as 15 years.

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