Working with Two Equations using Graphs

Date: 01/16/2002 at 17:17:01
From: Brenda
Subject: Working with Two Equations using graphs

I'm having major trouble with graphing some points using the formula
y = mx+b. The problem in my math book reads: 

You estimate that you can make muffins for $.30 each. Advertising will 
cost $18.00. You sell the muffins for $.75 each. Write and graph 
equations to represent income and expenses. Find the break-even point. 

I know the equations are y = .75x and y = .30x+18. I also know how to 
graph the line y = .75x, but I get really confused about how to graph 
the line y = .30x+18. So when I try to graph the expense line I get it 
confused with the income line. 

If you could help me, and tell me an easy way to remember how to do 
it, I would appreciate it greatly. 

Thank you for taking time to read this. :-)

Date: 01/17/2002 at 09:47:47
From: Doctor Ian
Subject: Re: Working with Two Equations using graphs

Hi Brenda,

You have the right equations. The first one is going to start at the 
point (0,18) and rise with a particular slope; the other will start at 
(0,0) and rise with a steeper slope, which is why it will eventually 
catch up. 

     |                     i     e
     |                   i   e
     |                 i e
     |               b        
     |           e i
     |       e   i
     |   e     i
     e       i 
     |     i
     |   i
     | i
     i------------------

One thing that might help you is to get used to looking at graphs as 
sources of information, rather than as a kind of puzzle that you have 
to learn to solve.

For example, by looking at the 'e' line in the graph above, we can see 
that there is some initial cost that has to be paid even if no muffins 
at all are sold. And because the graph is a line, we can see that each 
additional muffin made incurs a constant cost. 

Similarly, by looking at the 'i' line, we can see that if you don't 
sell any muffins, you don't make any money. And we can see that since 
the slope of the income line is steeper than the slope of the expense 
line, you'll eventually start making a profit if you can just sell 
enough muffins. (If the slope were less steep, this business would be 
in trouble - sort of like the old Internet startup joke: "We lose 
money on every sale, but we make it up in volume.")

A good habit to get into would be to take a few seconds to think about 
the meaning of _any_ graph that you see. After a while, it will become 
second nature, and it can prevent you from making silly mistakes. (If 
you work your way to a solution using algebra, and you get an answer 
that disagrees with the graph, that's a strong indication that you 
should check your work.) 

To draw a line on graph paper, you only need to find two points, 
because you can fill in the rest with a ruler. And when you have an 
equation like 

  y = 0.30x + 18

the two easiest points to come up with are the places where the line 
intersects the x- and y-axes. How do you find those? Well, when the 
line intersects the y-axis, the value of x must be zero:

  y = 0.30(x) + 18

    = 0.30(0) + 18

    = 18

So one point on the line is (0,18). And when the line intersects the 
x-axis, the value of y must be zero:

       y = 0.30x + 18

       0 = 0.30 + 18

  -0.30x = 18

       x = 18 / (-0.30)

Can I make a suggestion here? Whenever you're dealing with money, use 
pennies as your units instead of dollars:

       x = 1800 / -30

         = -60

So another point on the line is (-60,0). That's pretty far off to the 
left, so you may not want to extend your graph that far. So another 
good value to choose is x = 1:

       y = 30x + 1800

         = 30(1) + 1800

         = 1830

So now your two points are (0,1800) and (1,1830). From those two 
points, you can fill in the rest of the graph. 

If you're getting the two lines confused, a good trick to use is to 
graph them in different colors. Traditionally, red is used to indicate 
debt, and black is used to indicate profit, so you might make the 
expense line red and the income line black.  

That way, before the break-even point, the red line will be higher 
than the black line, putting you 'in the red' (i.e., you've spent more 
than you've brought in); and after the break-even point, the black 
line will be higher than the red line, putting you 'in the black' 
(i.e., you've taken in more than you've spent). 

But an even better trick, as I pointed out earlier, is to have a firm 
grasp of the _meaning_ of the graph. That way, you know what to 
expect: Two crossing lines, where initially the expenses will be on 
top, and later the income will be on top. Once you have that straight, 
it won't be as easy to become confused about which line represents 
what. 

Note that finding the break-even point is easier with algebra than 
with a graph.  You have two expressions for y:

  y = 75x

  y = 30x + 18

At the break-even point, the lines will intersect, which means they'll 
have the same value of y, which means the value of y will be equal, 
which means that for that particular value of x, it will be true that

  75x = 30x + 18

which is pretty easy to solve.  

So why bother to use graphs? Well, most real-life situations aren't 
nearly as simple as the one described by this problem! For a pair of 
lines, finding an algebraic solution is pretty easy. But as you add 
more equations, and as the equations start to wiggle (because they're 
polynomials, or exponential functions, or whatever), solving things 
algebraically becomes harder and harder, while finding graphical 
'solutions' remains pretty straightforward... assuming that you can 
generate the graphs.  

If you're confused about the whole concept of simultaneous linear 
equations, you might want to read this answer from the Dr. Math 
archives:

   The Idea behind Simultaneous Equations
   http://mathforum.org/dr.math/problems/laura.10.14.01.html   

Does this help?

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