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Simultaneous Equations and the Addition Method

Date: 06/01/98 at 00:16:57
From: sarah hamler
Subject: addition method(how do you do it|)

I'm trying to solve for x and y. Here's how I started:

   3x + 15y = 23
   8x + 14y = 70

I multiply the top by 14 and the bottom by -14:

   14(3x + 15y) = 23(14)
     42x + 210y = 322

   -14(8x + 14y) = 70(-14)
     -112 - 196  = -980

Now what?

Date: 06/01/98 at 01:07:33
From: Doctor Pat
Subject: Re: addition method(how do you do it|)


Relax, take a deep breath. You almost had it right. Here we go.

I usually decide right from the start if I want to get rid of x or y; 
that's important.

If I want to get rid of x, I will multiply each equation by the 
x-coefficient value of the other equation, making one of them 
negative. In your equation, I will multiply the top equation by 8 and 
the bottom by -3. The x-values will cancel out when we add them. 
    (8) * [3x + 15y = 23]  ------>   24x + 120y = 184 
   (-3) * [8x + 14y = 70]  ------>  -24x -  42y = -210
   Now just add down the like      -------------------- 
    terms columns to get              0x +  78y = -26   
Now solve 78y = -26 to get y = -1/3.     

Return to the original equations (either one) and use this value for 
y to solve for x:

   3x + 15y = 23
   3x + 15(-1/3) = 23 
   3x - 5 = 23
   3x = 28 
    x = 28/3 

Now we substitute both values into the other equation (the one we did 
not just use) to check our answers. This is important as it checks for 
any mistake along the way:

   8x + 14y = 70
   8(28/3) + 14(-1/3) = 70 
   224/3 + -14/3 = 70
   (224-14)/3 = 70  
   70 = 70  

Eureka, it works.  

You could have done the same thing eliminating the y terms first. To 
do that, we would just multiply the top equation by 14 and the bottom 
by -15, which is what you started to do. Go back and change the -14 to 
-15, and I think you'll end this problem with a smile on your face.  

I hope this helps a little. If you have more problems, write us again.  
Good luck.  
-Doctor Pat, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
Associated Topics:
High School Basic Algebra
High School Linear Equations

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