Algebra: re-arranging mathematical equations.

Let's return briefly to considering relationships among people, rather than numbers.

Suppose I tell you that the following statement is true:

"Trevor is Jen's brother"

If that's the case, then you'll probably agree that the following statement must also be true:

"Jen is Trevor's sister".

You can use the information in one statement to come up with, or derive, the other statement. As long as one is true, the other will be true. The Greek mathematicians spent a lot of time thinking about ways to derive true statements from other statements that they knew to be true. This sort of activity, which they called deduction, occurs frequently in math. It is a math skill that gets easier with practice.

Question: Can you think of a rule that people could follow to change statements like "Trevor is Jen's brother". into statements like "Jen is Trevor's sister?"
(Answer 1)

Question: If they follow this rule correctly, will the new statement always be true?. Are there any situations when the statement could be false?
(Answer 2)

In order to find the missing piece of Nov. 7 precipitation information in the second part of the precipitation scenario in section 3.10 (where coffee was spilled on the record), you relied on the fact that you knew the relationship between all of the numbers in the equation to deduce the missing piece of information.

You knew that if this mathematical statement was true:

”The total precipitation for the week can be found by adding up all of the precipitation amounts for each day of the week”

Then this mathematical statement must also be true:

”The amount of precipitation that fell on Nov. 7 will be equal to the difference between
the total precipitation for the whole week and all of the other weekly amounts of precipitation.”

(try reading that through a few times)

Or, to put it in terms of equations, you knew that if this mathematical equation was true:

day1 + day2 + day3 + day4 + day5 + day6 + day7 = total_precipitation

Then this mathematical statement must also be true:

day7 = total_precipitation – (day1 + day2 + day3 + day4 + day5 + day6)

You have deduced the truth of the second statement from the first.



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Answer 1: One possible answer would be to start with the first statement, switch the names around, and replace the word brother with sister.

Answer 2: Yes- the new statment will always be true. There's no situation where one person is another person's brother, but the other person is not the first person's sister.
 

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Copyright Jen Schellinck, 2006