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### Subtracting One Equation from Another

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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

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.
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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/
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Associated Topics:
High School Basic Algebra
High School Linear Equations
Middle School Algebra
Middle School Equations

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