systems of equations: elimination

[dragonstar57]

their is no specific problem I need help on but a operation we have used in my class frequently which I don't understand.

the elimination method for solving linear systems of equations

it has something to do with taking the coefficient of a variable in one equation and making it the opposite of the coefficient of the same variable in the other equation then eliminating (doping I guess) them from the equations.

can someone explain how this works and perhaps frame a few examples?

[ACUV]

Maybe this is what you are talking about, here are two simultaneous equations,

 

4H + 6T = 20

 

H + T = 4

 

H is a variable, in the first equation it has a coefficient of 4 and in the second equation it has a coefficient of 1.

Firstly, to get these coefficients the same many multiplication factors can be used but I will use a factor of 4 on the second equation.

The two equations now become the following two equations.

 

4H + 6T = 20

 

4H + 4T = 16

 

Then, another multiplication factor of -1 is applied to the second equation making them as follows,

 

4H + 6T = 20

 

-4H -4T = -16

 

The equations can now be added as follows,

 

add the H parts and add the T parts and let them be equal to the sum of the parts over the other side of the equals sign,

 

4H - 4H + 6T -4T = 20 - 16

 

2T = 4

 

T = 2

 

 

The original equations were,

 

4H + 6T = 20

 

H + T = 4

 

 

using H + T = 4 , if T = 2 then H = 2

 

using T = 2 and H = 2 does 4H + 6T = 20

 

4H = 8 and 6T = 12 , so 4H + 6T = 20

 

Then, H = 2 and T = 2 is a solution to the original equations.

 

edit, spelling

Edited October 15, 2012 by ACUV

[dragonstar57]

so you subtract one equation from the other?

[ACUV]

The idea is to make a coefficient of a particular variable in both equations equal numerically and opposite in sign.

 

When you add numbers like those following you get answers like those following,

 

a + - a = 0 , adding - a to a is the same as adding - 1 a to 1 a , the balance is 0 a = 0

 

b + - b = 0 , adding - b to b is the same as adding - 1 b to 1 b , the balance is 0 b = 0

 

- 3 c + 4 c = c

 

4 c + - 3 c = 4 c - 3 c = c

 

6 pq - 4 pq = 2 pq

 

-4 pq + 6 pq = 2 pq

 

......................................................................................................................................................................

 

H + 4 K = 20

 

-H - K = 10

 

The coefficients of H are the same in value and opposite in sign, so add the equations like the following,

 

H + ( - H ) + 4 K + ( -K ) = 20 + 10

 

H - H + 4 K - K = 30

 

3 K = 30

 

K = 10

 

Using, H + 4 K = 20

 

H + 40 = 20

 

H = -20

 

The original equations were,

 

H + 4 K = 20

 

-H - K = 10

 

H = -20 , K = 10 is one solution of these simultaneous equations.

 

...........................................................................................................................................................

 

S + K = 20

 

-S - 3K = 40

 

adding the equations,

 

-2K = 60

 

etc

 

......................................................................................................................................................

 

acuv

[mississippichem]

To the OP,

 

If you want to dig a bit deeper you should read about Gaussian elimination or Gauss-Jordan elimination.

 

Basically you can formulate systems of linear equations as a "vector of variables" (not always strictly a vector in terms of tensor transformation rules, but a column matrix anyway) multiplied by a matrix of coefficients set equal to another column matrix of constants. Gaussian and Gauss-Jordan elimination provide a number of operations you can use to solve the system in this matrix formulation. Multiplying a row by a constant and adding/subtracting it from another row is one such operation. This will typically be covered in the first few lectures of an undergraduate linear algebra course.

 

Not essential to your question but I think it would be a good thing to look up if you want to learn more about systems of linear equations.

Edited October 16, 2012 by mississippichem

[ACUV]

I assumed a basic level of manipulation of simultaneous equations. I hope it helps.

[mississippichem]

I assumed a basic level of manipulation of simultaneous equations. I hope it helps.

 

Your answer was more than adequate. I had no intention of belittling it. I was just pointing to further reading iff dragonstar57 happens to be interested.

 

I'm a bit of a linear algebra fanatic. Can't help it.

 

Edit:

 

Definition of mathematics stolen from my multivariable calculus professor:

 

mathematics-the study of reducing hard problems to linear algebra

Edited October 16, 2012 by mississippichem

[ACUV]

 

I had no intention of belittling it.

 

 

I did see the possibility in terminology used by the OP that other more advanced options should be considered. You have considered this and I myself am being pointed by you to an area of interest. Matrices was never something I had been taught. Thanks!

[mississippichem]

I did see the possibility in terminology used by the OP that other more advanced options should be considered. You have considered this and I myself am being pointed by you to an area of interest. Matrices was never something I had been taught. Thanks!

 

Glad I could help.

 

That's why I post here; the off chance that something I say might lead to someone else furthering their knowledge or the other way around, someone posting something that might further my education.

 

I declare this thread a success!