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Solving Systems of Linear Equations by Combination

Date: 11/11/2003 at 14:37:11
From: judy
Subject: solvin systems of linear equations in three variables

We are working on systems of equations problems that have three
equations and three variables.  Our text shows a method where
equations are added to each other.  How do you decide which equations
to add?  Also, they sometimes use a number to multiply an equation by.
I am totally confused on this subject.

Date: 11/11/2003 at 17:04:47
From: Doctor Riz
Subject: Re: solvin systems of linear equations in three variables

Hi Judy -

Thanks for writing.  The basic idea to keep in mind when working with 
systems of equations is that you want to reduce the number of 
variables.  When you have only a single variable in an equation, you 
can solve and determine what that variable represents.

For example, if you have x + 3 = 5, it's clear that x must equal 2.  
But if you have x + y = 5, there are lots of possible combinations of 
two numbers that add to make 5, such as (0,5) or (-1,6) or (2,3) and
so on.

So the goal is to take a system of equations and reduce it until you 
are left with an equation that has only one variable in it.  Once you 
determine the value of that one variable, you can substitute that
value into all the equations for that variable, and that will get you
on your way to solving for all the variables.

One common way to reduce the number of variables is to combine 
equations and cancel them out.  For example, suppose your system has 
these two equations:

  3x + y = 7
   x - y = 1 

If you just add those two equations together, the +y and -y will 
cancel, leaving you:

  3x + y = 7
   x - y = 1 
  4x     = 8
       x = 2

Now we know that x = 2, and when we substitute 2 in for x in either of
the original equations we then have an equation with only y as a
variable, and we can solve for y:

    3x + y = 7        OR        x - y = 1
  3(2) + y = 7                  2 - y = 1
     6 + y = 7                    - y = -1
         y = 1                      y = 1

We have an answer that works in both equations, with x = 2 and y = 1.

Now suppose the two equations had been:

  3x + y = 7
   x + y = 5

This time if we add them together nothing will cancel out, so that 
doesn't help us.  But what if we multiplied the second equation by -1 
so that the +y became -y?  Then it would cancel out when we added it
to the first equation.  Let's try that:

  3x + y = 7   leave this one alone  -->  3x + y = 7
   x + y = 5   multiply this by -1   -->  -x - y = -5

Now when we add them the y's cancel again and we get:

  3x + y = 7
  -x - y = -5
  2x     = 2
       x = 1

Once we know x = 1, substitute that in for either x in the original 

  x + y = 5      OR      3x + y = 7
  1 + y = 5            3(1) + y = 7
      y = 4               3 + y = 7
                              y = 4

So (x,y) = (1,4) is our solution.

Note that we could also have subtracted the two original equations,
which would have done the same thing as multiplying by -1 and adding.
But I've found that most students find it easier to add than subtract, 
so it's often safest to modify the equations so a variable will cancel 
when they are added.

Sometimes you need to multiply by numbers as well, but the goal is 
still to create a system where one variable will cancel out when you 
add the equations.  Suppose you have:

  7x + 2y = 1
  5x - 3y = -17

If we add now, nothing cancels for x or y.  So let's do some renaming.  
Since 2 and 3, the coefficients of y, can both be easily turned into
6, multiply each equation by whatever you need to make the y
coefficents be 6:

  7x + 2y = 1    multiply by 3 -->  21x + 6y = 3
  5x - 3y = -17  multiply by 2 -->  10x - 6y = -34

Now when we add the y's cancel out again, and we have:

  21x + 6y = 3
  10x - 6y = -34
  31x      = -31
         x = -1  and you can go back with x = -1 and solve for y.

So the idea here is that when you start with two equations and two 
variables, you want to combine the equations to leave one equation 
and one variable.  This is called "reducing the system".

Now, the same approach works with a 3-equation, 3-variable sytem.  The 
first thing we want to do is reduce the system to 2 equations and 2 
variables.  Then it becomes just like the problems we just did, and 
once you've figured out the values of those 2 variables, you can go 
back and substitute them into one of the original equations to find
the 3rd variable.

So how do we reduce from a 3 equation/variable system to a 2 equation/
variable system?  Use the same method we just looked at.  Suppose your 
3 variables are a, b and c.  If you combine the first two equations
and eliminate c, you will be left with an equation having only 'a' and
'b'.  Then combine the second and third or first and third equations
and again eliminate c.  Now you have a second equation with only 'a'
and 'b'.  So now it looks like what we just did--you have two 
equations with two variables, a and b.  Combine those two again and
eliminate 'a' or 'b' and now you have one equation with one variable.

Solve for that first variable, then plug it into one of the 2-variable 
equations and solve for the 2nd variable.  Then take both those values 
and plug them into one of the original equations and solve for the 3rd 

So in essence, any system of equations problem is done by reducing the 
number of equations and variables using combination and elimination.  
Once you have reduced the system down to one equation and one
variable, you then work your way back out, solving for specific values
of the variables one at a time.

As for how to choose the best equations to combine, that's really a 
matter of experience.  Look for coefficients that can easily be 
matched, like we did when we turned 2 and 3 both into 6.  Remember
that when you multiply through the equations, each piece gets multiplied. 
Keep in mind also that you want to wind up with both a positive and 
negative version of the number (like we did when we got +6 and -6) so 
that they will cancel when you add.

I hope this helps--systems problems are actually kind of fun once you 
figure out how to do them.  I used to give my kids a 6 equation-6 
variable system as a bonus question, and as long as you go through the 
steps of first reducing it to 5 equations and variables, then 4, then
3 and so on, you can figure them all out.

By the way, there is another common method used in systems problems 
called substitution, but since you specifically asked about combining 
equations I only talked about that.

Good, luck, and write back if you are still confused.

- Doctor Riz, The Math Forum 
Associated Topics:
High School Linear Equations

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