Linear Algebra and the aMAZEing Math Maze

013B

Hello - welcome to Linear Algebra.  First we will discuss the answer to #13a in the aMAZEing Math Maze, then we'll ask question #13b and finally we'll talk bit about Linear Algebra specifically.  

Question 13a

As you may recall, we had Bob running around with a jar trying to catch lightning bugs.  Bob caught 7, Pete had 18.  Bob was catching 2 per minute Pete 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?

If we look at Bob first, He starts with 7 and adds 2 to his total every minute.  So, his equation would be      7 + 2t = x.  Where t is the amount of time (every minute is 2 more lightning bugs), and x is the total number of lightning bugs caught.  Using the same variables, Pete's equation would be 18 + t = x.  Pete starts with 18 and adds 1 every minute or every t.  We want to know when the x's in each equation are equal.  That will happen when they each have caught the same number of lightning bugs.  

Since 7 + 2t = x and 18 + t = x we can say that 7 + 2t = 18 + t (We're substituting 18 + t for x).  Solving for t we want to subtract 7 from both sides of the equation.  On the left 7 - 7 + 2t leaves us with 2t and that equals 18 - 7 + t.  Now our equation is 2t = 11 + t.  Subtract t from both sides.  This gives us t = 11.

t = 11 or - the answer is B) 11 minutes

If you graphed both equations, they would intersect at t = 11.

Question 13b

Bob has $100 to spend on the movies.  He spends $20 every time he goes. Jim has $50 but his cousin owns the theater so he only spends $10 when he goes.  Assuming they both go to the movies the same amount of times.  How many movies will they see before they both have the same amount of money?

a)  20 movies

b)  10 movies

c) 5 movies

d) -5 movies

Linear Algebra

Linear algebra is the study of straight lines and the formulas that make them.  All linear equations can be put into slope intersect form, which is y = mx + b.  In this equation y is the dependent variable (meaning it's value depends on the value of x).  The m stands for slope (we're not sure why, the most convincing reason I've seen is that the m stands for modulus as in "modulus of slope.")  Of course, this begs the question what does modulus mean, I'll let you use the dictionary for that one.

Everyone seems to worry about the m, so not many ask why the b stands for the y intercept.  When x is 0 the graphed line will intersect the y axis at the y intercept.  We use a b for the y intercept so we can use an a for the x intercept (where the line would intersect the x axis when y is 0.

There are a couple of ways to determine the linear equation of a line.

Two Points - If you have coordinates (X, Y) and (x, y) then m = (Y - y)/(X - x)  Now that you have the slope in y = mx + b, to find the b put in either set of coordinates and solve for b.  (I don't know how to do subscripts without resorting to HTML - so, I'll differentiate the points by using caps and lower case x and y).

On Point and the Slope - Even easier, you put in the coordinates of the point for x and y and solve for b.

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