Subtracting One Equation from Another

Date: 05/30/2001 at 23:18:03
From: Lee
Subject: Subtracting one equation from another

I'm 52.  No one ever answered this question in school. I understand 
that if you have two equations with two variables - the same variables 
in each equation, say X and Y - there are two generally accepted ways 
of solving the equations. You can isolate one variable in one 
equation, then substitute it in the second equation. Or you can 
manipulate one of the equations so that you have equal values of one 
variable, then subtract one equation from the other, eliminating the 
variable. I can do the subtraction and solve the equations, but no one 
ever answered this question:

Why can you subtract one equation from another? Not, why can you 
subtract one mixed variable from another, for example, 5X - 3X or 
9X squared - 5X squared (I'm squaring the X). That's clear enough.  
But why can you subtract one equation from another and get something 
meaningful that you can use to solve the equations? The explanations 
I've heard are just given in more math jargon, which is no help.

A side question. If you graph the equations, you get two lines. How 
can you subtract a line from another line? You can subtract the length 
of one line from the length of another, but how can you subtract 
lines? Lines are not numbers; they're marks on pieces of paper or 
chalkboards. There are points all along the lines, but no one says, 
"Subtract every point on the second line from every point on the first 
line."  I still don't get it.

Date: 05/31/2001 at 09:28:05
From: Doctor Peterson
Subject: Re: Subtracting one equation from another

Hi, Lee. Great question!

Let's do an example, just to have a specific example to discuss. We 
want to solve

    2x + 5y = 26
     x - 3y = 2

We double the second equation, giving

    2x - 6y = 4

and then subtract the equations to get

    0 + 11y = 22

so that y = 2.

Why can we subtract one equation from another? If you picture an 
equation as a balanced scale, this is easy to explain: take the same 
amount off both sides of the scale, and it remains balanced. Here, 
"2x - 6y" is the same as 4; so we can subtract 4 from each side of 
the first equation (which will not change its meaning) by subtracting 
2x-6y from the left side and 4 from the right. So the new equation 
will be true whenever both original equations are true. This gives us 
a simpler equation we can solve in order to find the solution to the 
original pair.

Technically, we want to transform the problem into an equivalent one, 
that is, one with exactly the same set of equations; but the new 
equation by itself is not equivalent to both original equations: 
although any solution to the original pair will make this equation 
true, not all solutions to the new equation will be solutions of the 
pair. You can see this easily here, where x doesn't even appear in the 
new equation, so it can be anything. What's really happening is that 
the new equation _together with_ one of the pair is equivalent to the 
original pair.

Now, what does this mean in terms of the graphs of the equations? Not 
much. Notice that if you multiply an equation by 2, as we might do 
before subtracting, we haven't changed its meaning at all - it's still 
the same line. So this is an action on the equation that has no effect 
on the graph. Since we can subtract any such form of the equation of 
one line from the other, you can see that the line itself doesn't 
determine what will happen when we subtract equations. I can't draw 
two lines and then say "this third line is the difference between 
those two lines," because there are many different possibilities for 
the result, depending on what equation we used for each line. In other 
words, you can't talk about subtracting two lines in this sense; you 
can only subtract equations.

But you might like to try writing several equivalent equations for 
each of two lines, and then subtracting various pairs of equations and 
graphing them. You'll find that all the resulting lines have one thing 
in common: they all intersect at one point (assuming the original 
lines intersected). Two of these are used in solving the system: one 
that is horizontal, and gives us y, and one that is vertical, and 
gives us x.

Now let's do one more thing: see how the two methods, subtraction and 
substitution, are equivalent. If we solved the same pair of equations 
by substitution, we might do this:

    2x + 5y = 26
     x - 3y = 2

From the second equation, x = 2 + 3y. Substituting in the first,

    2(2 + 3y) + 5y = 26

Now, rather than solve this, I'm going to rearrange it to show that 
this is the same as subtracting twice the second equation from the 
first:

    2(3y) + 5y = 26 - 2(2)

    5y - 2(-3y) = 26 - 2(2)

The x's have disappeared, which is our goal in either method; the left 
side is the 5y from the first equation minus twice the -3y from the 
second equation; and the right side is 26 minus twice the 2 from the 
second equation. So you could say that this method is just a different 
way to substitute.

Let me know if you need any more help with these ideas.

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