Linear Systems

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Aug 19, 2019
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The combined ages of a mother, her son, and her daughter are 61 years. The mother is 22 years older than her son and 31 years older than her daughter. How old is each person. Solve using a linear system.
 
It helps to develop a sense of abstraction right from the start. You can do this by writing definitions. Make simpler things represent more complicated things.

Mother's Age - for example, might be just M.
Her Son's Age might be just S.
Her Daughter's Age might be just D.

The abstraction, of course, is to think of M, S, and D as a number that you actually know - or maybe have a little faith that you will know it. This gives you permission to deal with it.

How old is Mom? M - of course. What is the current value of M? This is where the faith comes in. The definition is good enough for now.

After definitions, one must learn to translate.

"The combined ages of a mother, her son, and her daughter"

What does that mean? From your definitions, you need to see that this is M + S + D. We simply combined all the ages with addition! The "addition" may not have been obvious, but we know it's a linear system, so it's less likely that it will be more complicated than that.

Where does that leave us?
 
So, what is the actual formula for solving?
That's the part you're supposed to be creating from your brain. It requires that tiny bit of abstraction I was talking about. Words mean things. In this case, we need to find a way to figure out what they mean.
 
"The combined ages of a mother, her son, and her daughter are 61 years."
Using "M", "S", and "D" to represent the ages of the mother, son, and daughter, respectively, there "combined ages" is the sum, M+ S+ D. Thar is "61 years" so M+ S+ D= 61.

"The mother is 22 years older than her son"
So M= S+ 22

"and 31 years older than her daughter."
M= D+ 31

That's where we get the equations. It is not a matter of a "formula", it is a matter of thinking!
 
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