Linear

013A

Well, you are here because you searched for linear, or because you missed question 13 of the aMAZEing Math Maze.  Either way, this is a good place to be.  Linear refers to an equation that plots out to a straight line.  Another way to say this is a first degree equation.  If you really want a definition - you could check the AudioEnglish.org site.  

In this context, we are working with the definition that concerns itself with equations.  Question 13 was one of those dreaded word problems.  It's starts out saying that Bob has $200 and is saving an additional $20 a month.  

So, the equation starts out with Bob = $200.  Bob is much more than his money, but this problem only concerns itself with Bob's money.  We could make it x = $200 if you prefer, but I think Bob makes a more descriptive variable name.  Every month Bob adds an additional $20 to the account.  So the first month he adds $20 and the second month he adds another $20 for a total of $40 added to the $200 - and so forth.  The equation now becomes Bob = $200 + $20m.  Where m stands for the variable month.

Now, Jane starts out with $400 but she spends $30 every month.  Her situation is similar to Bob's except Jane is spending money where Bob is saving it.  Jane = $400 - $30m.  We subtract $30 from her account every month.

The question asks "how many months before they have the same amount of money in their savings accounts?"  Another way of phrasing this would be - how many months until Bob = Jane?  Well Bob is $300 + $20m and Jane is $400 - $30m, if Bob = Jane then $200 + $20m = $400 - $30m. Aha!!  we no longer have a word problem - now we just solve for m!

Step #1 - add $30m to both sides of the equation.  Remember, we can do pretty much anything as long as we do it to both sides.  We are doing this to get the variable all on one side of the equation.  We end up with $200 + $20m + $30m = $400 - $30m + $30m.  Adding like terms leaves us with $200 + $50m = $400.

Step #2 - subtract $300 from both sides.  We do this to get the variable all by itself.  $200 -  $200 + $50 = $400 - $200.  Combining like terms gives us $50m = $200.

Step #3 - Well, m still isn't by itself.  It's currently being multiplied by $50.  So divide both sides by $50 to see what m equals.  $50m/$50 = $200/$50.  On the left side, $50/$50 is 1, so we end up with m, by itself (which is what we wanted).  On the right side $200/$50 = 4.  So, our equation is m = 4. Which is answer A).

Too much work!

O'k, it's a bit much I admit.  Hopefully you can understand it well enough to skip some of those steps.  An alternative method would have been to look at the possible answers.  The first answer was 4, so does 200 + 20(4) = 400 - 30(4)?  Turns out it does, but if it didn't then you could try  the second answer option.

Why is it Linear, and can I get another shot at this puppy?

And before you say it - no, we are definitely not advocating that people shoot at puppies!  It's an expression.

If you plot these equations, they will show two intersecting linear lines.  Thus the topic Linear (besides, I already had a page titled Word Problem (although it occurs to me that we could have titled this Word Problems - oh well, that will probably come up later).

Question 13a  Bob is catching lightning bugs.  He has caught 7 of them when he meets his friend Pete who has caught 18.  Bob has been catching lightning bugs at the rate of 2 per minute while Pete only catches one per minute.  Assuming they keep catching lightning bugs at the same rate, how many minutes until Bob has the same number as Pete?

A)  4 minutes

B)  11 minutes

C)  25 minutes

D)  Give it up Bob, you'll never be as good as Pete!


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