Solving Equations in Two Variables by Addition/Subtraction

Date: 09/08/2001 at 16:43:06
From: nasha
Subject: How to solve system equations in two variables by addition 
and subtraction

The problem is 

       2x + 3y = 7
       4x - 5y = 25

I have no idea how to solve this problem at all. Please help me; I'm 
really stuck.

Date: 09/09/2001 at 00:00:29
From: Doctor Ian
Subject: Re: How to solve system equations in two variables by 
addition and subtraction

Hi Nasha,

Suppose for a moment that we are trying to solve a slightly different 
problem:

  2x + 3y =  7
  2x + 4y = 10

Just looking at the two equations, we can see that an extra y on the 
left side corresponds to an extra 3 on the right side. So we know that 
y = 3 without having to do anything. 

If instead we have

  2x + 3y =  7
  2x - 4y = 21

In this case, see that a loss of 7y on the left (from + 3y to - 4y) 
corresponds to a gain of 14 on the right. So we know that -7y = 14, or 
y = -2.

By now, you might be wishing that you had to solve a problem like 
this, instead of the one that you were given. But wait! Is there a way 
that you can turn your problem into a problem like this? 

What if you multiply everything in the first equation by 2? Then you 
would have

  4x + 6y = 14
  4x - 5y = 25

Can you take it from here? 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions.   

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