Missing Factors

Date: 01/29/98 at 21:27:30
From: Cindy Bayliss
Subject: Algebra

Help! I can not figure out how to find the two missing factors when 
given two equations. I know how to use the addition method and the 
multiplication method but what do you do when the left number doesn't 
go into the right number evenly, such as in these two problems:

3x + 4y = 1
2x + 2y = 2

  x - 6y = 2
2x -   y = 17

I have tried but I just can't get it. Please help.

Thanks,
Cindy

Date: 01/29/98 at 22:00:19
From: Doctor Gary
Subject: Re: Algebra

One of the great things about an equation is that you can always do 
the same thing to both sides, without disturbing the relationship of 
equality between the left and right sides of the equation.

Suppose, for example, we knew that:

          3x + 4y  =  1

and also that 

          2x + 2y  =  2

Remember "least common multiples"? You can use the same principle here 
to discover what factors should be used to multiply both sides of each 
equation.

The lowest common multiple for 3x and 2x is 6x.  We can obtain two 
equations with a term of 6x by:

        multiplying 2 times both sides of 3x + 4y  =  1,  and
        multiplying 3 times both sides of 2x + 2y  =  2

Now our two equations are:

          6x + 8y  =  2
          6x + 6y  =  6

We can subtract "the same thing" from each side of the top equation.  
Since the second equation tells us that (6x + 6y) is "the same thing" 
as 6, we can subtract (6x + 6y) from the lefthand side of the first 
equation and 6 from the righthand side. This tells us that 2y is the 
same as -4. In other words, y is -2. We can now "plug" this value of y 
into either of our original equations, to see what x is.

      3x + 4(-2)  =  1

          3x - 8  =  1

              3x  =  9

               x  =  3

But don't take my word for it. Try those values for x and y in the 
other original equation:

    2(3) + 2(-2)  =  2

           6 - 4  =  2

> x - 6y =  2
>2x -  y = 17
>
>I have tried but I just can't get it. Please help.


What did you try?

My favorite advice about math is not to do anything that doesn't make 
a whole lot of sense. Math is very easy if you become familiar with 
and comfortable applying just a few basic principles.

-Doctor Gary,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/