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Un-learning Unknowns

Date: 10/12/2010 at 17:59:04
From: Sam
Subject: Confusion about systems of equations application

Today, we started a unit on linear programming, but we first had a review
of systems of equations, specifically for the upcoming SAT. We were
looking at this problem:

Each time Sue rents a bicycle, she pays a fixed base cost plus an hourly
rate for the time the bicycle is rented. Last Saturday she paid $12.00 to
rent a bicycle for 6 hours, and yesterday she paid $9.50 to rent a bicycle
for 4 hours. Which of the following equations shows the total cost C, in
dollars, for Sue to rent a bicycle for n hours?

   a) C = 1.25n
   b) C = 1.25n + 4.50
   c) C = 2.00n + 2.50
   d) C = 2.50n + 2.00
   e) C = 4.50n + 1.25

First, I said ...

   12.00 = m(6) + b

... where m is the slope of the line, or rate; and b is the fixed base
price, or the y-intercept. Then, I said

    9.50 = m(4) + b.

Six and 4 are the number of hours. For the first equation, I set ...

       b = 12 - 6m 

... and I substituted in this value into the second equation, so I got

    9.50 = 4m + (12 - 6m).
   
I then combined like terms and solved for m, arriving at m = 1.25. This
made b = 4.50. Therefore, answer b) must be right.

But then I stepped back and thought for a moment. I have been doing
systems of equations for so long that I have simply been going through the
motions. As I looked over my work, I became more and more confused.

Please correct me if I am wrong, but I have always thought of systems of
two variable linear equations as lines to plot on a graph. The point where
the lines intersect is the pair of coordinates that works for both graphs,
and this is the solution to the system. For example, in y = 5x and 
y = -3x + 2, you are assuming that y and x are the same value, and this
allows you to then substitute or eliminate. (It would be very helpful if
you could comment or help me with this statement I just made, for I never
fully understood how you can substitute y or x values if the equations are
different.)

To solve this problem, I set up two equations. One was 12.00 = 6m + b and
the other was 9.50 = 4m + b. My main confusion comes from the fact that a
system of equations is the point where the two lines intersect. But how
can you plot the lines here and see where they intersect? There is no y or
x value, and you do not know m or b.

My second question came up when I wondered about plotting the lines. I
visualized some lines, but I thought they were the same. What I am trying
to say is that the two equations ...

   12.00 = 6m + b
    9.50 = 4m + b

... already have the same y-intercept and the same slope. They have
different x-values, but the y-values correspond in a way that is
proportional. If the two lines are the same, then how can you have an
intersection point?

Essentially, the y-intercept and the slope of a line are the defining
qualities of a line. (Right? Please help me with this.) If that is true,
and we have two equations with the same slope and y-intercept, then how
come they are not the same line? I mean ...

     126 = y + 4x
      96 = y + x 

... are different because of the slope and y-intercept, right?

I am a slow learner, but I understand stuff when it is detailed, so please
thoroughly explain the answer. THANK YOU SO MUCH because I am really
confused right now.



Date: 10/12/2010 at 22:56:51
From: Doctor Peterson
Subject: Re: Confusion about systems of equations application

Hi, Sam.

Let me start by reassuring you that you did get the right answer. Now, on
to your questions.

>But then I stepped back and thought for a moment. I have been doing
>systems of equations for so long that I have simply been going through
>the motions. As I looked over my work, I became more and more confused.

Thinking about the meaning of what you're doing is a good thing!

> Please correct me if I am wrong, but I have always thought of systems of
> two variable linear equations as lines to plot on a graph. The point
> where the lines intersect is the pair of coordinates that works for both
> graphs, and this is the solution to the system. For example, in y = 5x
> and y = -3x + 2, you are assuming that y and x are the same value, and
> this allows you to then substitute or eliminate. (It would be very
> helpful if you could comment or help me with this statement I just made,
> for I never fully understood how you can substitute y or x values if the
> equations are different.)

That's exactly it: you're assuming both equations are true (since you're
assuming x and y are the solutions), so you can replace either side of one
equation with the other side. That's what "equal" means -- they're the
same value. So whichever you use, you get the same result.

> To solve this problem, I set up two equations. One was 12.00 = 6m + b
> and the other was 9.50 = 4m + b. My main confusion comes from the fact
> that a system of equations is the point where the two lines intersect.
> But how can you plot the lines here and see where they intersect? There
> is no y or x value, and you do not know m or b.

We tend to use x and y so much that we forget that you can use ANY
variables for the unknowns! In your case, the unknowns that you are
looking for are not x and y, but m and b. So if you were to plot the
lines, the axes would be labeled m and b, not x and y.

> My second question came up when I wondered about plotting the lines. I
> visualized some lines, but I thought they were the same. What I am
> trying to say is that the two equations ...
> 
>    12.00 = 6m + b 9.50 = 4m + b
> 
> ... already have the same y-intercept and the same slope. They have
> different x-values, but the y-values correspond in a way that is
> proportional. If the two lines are the same, then how can you have an
> intersection point?

Since m and b are now the unknowns, they are not the slope and intercept
of the lines; they are the variables! The lines are ...

   6m + b = 12.00
   4m + b = 9.50

... which you can think of as if they were

   6x + y = 12.00
   4x + y = 9.50

Each has its own slope (-6 and -4, respectively), and its own y-intercept
(12 and 9.5, respectively) in terms of the variables m and b.

> Essentially, the y-intercept and the slope of a line are the defining
> qualities of a line. (Right? Please help me with this.) If that is true,
> and we have two equations with the same slope and y-intercept, then how
> come they are not the same line? I mean ...
> 
>      126 = y + 4x
>       96 = y + x 
> 
> ... are different because of the slope and y-intercept, right?
> 
> I am a slow learner, but I understand stuff when it is detailed, so
> please thoroughly explain the answer. THANK YOU SO MUCH because I am
> really confused right now.

This idea of taking the slope and the intercept from the original problem
and turning them into the variables in a new problem (the system of
equations) trips up a lot of people. Let algebra help you see abstractly:
it doesn't matter what you call the variables in order to get this
completely. Some textbooks, in fact, use a lot of unknowns other than x
and y precisely so that students don't get too used to recognizing a kind
of equation by the letters, but instead learn to understand them by the
relationships they involve.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/12/2010 at 23:54:44
From: Sam
Subject: Confusion about systems of equations application

Dr. Peterson,

Thank you for your answer, but I need a bit of clarification. 

First, I am still a bit wary of the idea of changing the variables and
equations. Do you think you could give me some examples? If so, that would
be great.

Along with the problem I gave, when you say ...

    6m + b = 12.00 
    4m + b = 9.50

... do you mean that the x-axis is the slope value and the y-axis is the
y-intercept value? That would mean that a change in slope will cause a
change in the y-intercept. I do not think this is correct, for how could
you add the 6 times the slope if you already have a slope on the x-axis?

Also, with your example, how can you change the x-value that was given to
you as the slope? 

In addition, why are the lines ...

    12.00 = 6m + b
     9.50 = 4m + b

... not the same line? They both have the same slope and y-intercept.

I'm still confused, but I see where you might be going, and I thank you
for your patience and help. Thank you for all you do at Dr. Math!!

--Sam



Date: 10/13/2010 at 10:16:52
From: Doctor Peterson
Subject: Re: Confusion about systems of equations application

Hi, Sam.

As I said, this is a big hurdle for students, so I'm not at all surprised
that you still have problems with it! Let's try a different approach.

Here is a system of equations; can you solve it?

  5a + 3b = 9
  3a - 2b = 13

Of course you can. And when you do, you do not care at all whether I used
a and b, or x and y, or some other pair of variables. The names don't
matter; an equation is all about the relationship between the variables,
regardless of their names or meanings. That is, it is abstract, and not
tied to what the variables are.

Let me know if you're okay so far, and I'll continue answering your
specific questions after that.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/13/2010 at 15:48:39
From: Sam
Subject: Confusion about systems of equations application

Thank you for your response.

I think I am OK with that so far, but would you have to denote which
variable would go on the x axis and which variable would go on the y axis?



Date: 10/13/2010 at 23:19:43
From: Doctor Peterson
Subject: Re: Confusion about systems of equations application

Hi, Sam.

You COULD choose to graph the equations if you wish -- but the problem
doesn't necessarily involve graphing. That is just one way to
represent it. The problem itself is just about finding a pair of
values that satisfies both equations.

And if you did graph them, you could chose to put either variable on
either axis. There is nothing inherent in the problem that makes one a
better choice for the "independent variable" over the other.

In other words, graphing is one TOOL you can use to interpret the
problem -- namely, as the intersection of two lines. But the problem
itself doesn't require that tool, either for solving it or for
understanding it.

Now let's look back at your problem.

You are trying to answer this question:

   Each time Sue rents a bicycle, she pays a fixed base cost plus an
   hourly rate for the time the bicycle is rented. Last Saturday she
   paid $12.00 to rent a bicycle for 6 hours, and yesterday she paid
   $9.50 to rent a bicycle for 4 hours. Which of the following
   equations shows the total cost C, in dollars, for Sue to rent a
   bicycle for n hours?

That problem is not about a graph, but about rental costs. Your
approach to solving it was to represent it as a problem about finding
the equation of a line by looking for the slope and intercept. (There
are other ways to model the problem, but you chose this because it is
familiar. A standard method of problem solving is to model a new
problem as one that is familiar, so you can use tools you already have.)

You also recognized that "a fixed base cost plus an hourly rate" means
that the cost is a linear function. Suppose that function is

   C = mn + b

or, using the variables you are more familiar with,

   y = mx + b

Again, you think of it this way just because you want to turn the
problem into one you know how to solve. All these variable names -- x,
y, m, and b -- are things you chose to introduce, along with the idea of
graphing. (By the way, the variables m and b here are parameters,
which means they are considered fixed for the sake of the problem,
namely graphing a line, whereas x and y are considered to actually
vary. But really they are all just variables -- letters representing
numbers you don't know. They just play different roles.)

Now your problem becomes one of FINDING the parameters m and b, on the
basis of two data points, namely that (x, y) = (6, 12) and (4, 9.5)
are known to be points on the line y = mx + b. So you can write two
equations by replacing x and y with those pairs:

    12 = 6m + b
   9.5 = 4m + b

This changes the models. Where a moment ago we had m and b as parameters
describing a line (with x and y the variables), now they are unknown
quantities you want to find! So you temporarily forget the original
meaning of m and b, and just think of this as a new problem of solving
a pair of equations.

It would look more typical, from your experience, if the equations
were ...

   6x + y = 12
   4x + y = 9.5

... or ...

   x + 6y = 12
   x + 4y = 9.5

... depending on which of m and b you think of as "x." But it really
makes no difference. The point is that m and b are now just unknown
numbers we are trying to find. THEY ARE NOT A SLOPE AND AN INTERCEPT
at this point; in particular, they are not the slope and intercept of
these two lines! In fact, the slope of the line ...

   6x + y = 12,   or  y = -6x + 12

... is -6. But that doesn't matter, because we don't even have to ask that
question unless we choose to solve this system of equations by
graphing. We just follow the usual methods to solve the equation and
find what m and b are (namely 1.25, and 4.5). THEN we go back to the
equation where we first used them, and put these values in their places:

   C = mn + b

becomes

   C = 1.25n + 4.5

Now let's answer the questions you asked last time:

> Along with the problem I gave, when you say ...
> 
>     6m + b = 12.00 
>     4m + b = 9.50
> 
> ... do you mean that the x-axis is the slope value and the y-axis is the
> y-intercept value? That would mean that a change in slope will cause a
> change in the y-intercept. I do not think this is correct, for how could
> you add the 6 times the slope if you already have a slope on the x-axis?

If you chose to graph the equation 6m + b = 12, putting m on the
horizontal axis and b of the vertical axis, then the graph would indeed
show how the slope OF THE ORIGINAL EQUATION y = mx + b would force the
y-intercept to change in order for the line to pass through (6, 12).
That's because this equation represents the constraint that the line must
pass through this point; if you imagine the line as a straight stick
pinned to the plane at that point, and rotated that stick (changing the
slope), it would change where it crossed the y-axis. For example, if
m were 0, then the line would be y = 0x + 12 (so that y = 0*6 + 12 = 12
when x = 6), and so on.

But in solving the problem, you never stop to think about this level of
meaning (and it doesn't matter) because at the point where you are working
with the equation 6m + b = 12, m and b are nothing more than two numbers
you are trying to find. The problem has been abstracted from its original
context.

This is the power of algebra: we can turn any sort of problem into a
problem that we can solve without having to think about the meaning of the
quantities involved.

You also asked:

> In addition, why are the lines ...
> 
>     12.00 = 6m + b
>     9.50  = 4m + b
> 
> ... not the same line? They both have the same slope and y-intercept.

As I've said this time, m and b are NOT the slope and y-intercept of
either of these linear equations; they WERE the slope and y-intercept of a
different line, y = mx + b or C = mn + b. But for the purposes of solving
for m and b, they are just the two variables in this system of equations.
We don't think of them as slope and intercept at this point, because what
they were originally is irrelevant to the work we are doing. They are the
slope and intercept of a line we are not currently paying any attention to.

Does that help at all?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
  


Date: 10/14/2010 at 23:04:36
From: Sam
Subject: Thank you (Confusion about systems of equations application)

Dear Dr. Peterson,

I want to say thank you for all of your helpful responses. They really
helped me understand the material and I appreciate the patience and depth
of knowledge you have, along with all the other voluteers at Dr. Math!

Thank you 
--Sam
Associated Topics:
High School Linear Equations

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