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August 26th, 2011, 01:39 AM   #1
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Simultaneous Equations

Referring to Definition of Simultaneous equations

http://en.wikipedia.org/wiki/Simultaneous_equations

For example,

x+y=3
x-y=5

The Above case has two variables x & y -> 2

Answer:

x=4
y=-1

Question 1:

Since Simultaneous equations involves multiple variables, what are there any rules for creating Simultaneous equations?

i.e Are these valid examples?

1.

x+y+z=3
x-y-z=4

The above involves 3 variables, since there are three variables do we have to have 3 Simultaneous equations to calculate value of x,y & z?
i.e say one of equation to be added

say 2x+2y-5z=1

2.

a+b+c+d=4
2a-b+c-2d=5

The above involves 4 variables, The above involves 3 variables, since there are four variables do we have to have 4 Simultaneous equations to calculate value of x,y & z?

say 3a+2b+2c-d=1
5a+3b-3c+3d=3

Question 2:

Can I have both Upper and Lower case Letters in Simultaneous Equations?

Are these valid examples?

Example 1:

x + X =4
x - X =6

Example 2:

x + Y = 1
x - Y = 7

Question 3:

Since there are 26 Upper case characters A.... Z and 26 Lowe case characters a ... z which gives us a count of Total 52 variables, can we construct Simultaneous equations as an example to compute all values of the 52 variables?

So do we have to write 52 Simultaneous equations in this case?

Thanks & Regards,
Prashant S Akerkar
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August 26th, 2011, 04:33 AM   #2
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Re: Simultaneous Equations

Quote:
Originally Posted by prashantakerkar
Referring to Definition of Simultaneous equations

http://en.wikipedia.org/wiki/Simultaneous_equations

For example,

x+y=3
x-y=5

The Above case has two variables x & y -> 2

Answer:

x=4
y=-1
Quote:
Yes, 4+(-1)= 3 and 4-(-1)= 5.

[quote:1j2xmt2o]Question 1:

Since Simultaneous equations involves multiple variables, what are there any rules for creating Simultaneous equations?

i.e Are these valid examples?

1.

x+y+z=3
x-y-z=4
Yes, those are perfectly valid simultaneous linear equations. "Simultaneous equations" are simply two or more equations that must be satisfied by the same values for the variables. They do not necessarily have to be linear.

[quote:1j2xmt2o]The above involves 3 variables, since there are three variables do we have to have 3 Simultaneous equations to calculate value of x,y & z?
To get a unique solution, yes, you need as many (independent) equations as you have variables. Here, we could add the two equations to get 2x= 7 so that x= 7/2. Putting that into the first equation gives 7/2+ y+ z= 3 or y+ z= 3-7/2= -1/2. Putting x= 7/2 into the second equation gives 7/2- y- z= 4 or -y- z= 4- 7/2= 1/2 so that y+ z= 1/2 again. Since those are the same equation now, the best we can do is write y= 1/2- z. We can choose any value for z and get a different solution for each different z. We would say the set of solutions has "one degree of freedom" or "is one dimensional".

Quote:
i.e say one of equation to be added

say 2x+2y-5z=1
Okay, as I said before, the first two equations give x= 7/2 and y+ z= 1/2. Putting x= 7/2 into this new equation gives 7+ 2y- 5z= 1 or 2y- 5z= 1- 7= -6. If we multiply y+ z= 1/2 by 5 we get 5y+ 5z= 5/2. Adding that to 2y- 5z= -6, we have 7y= -6+ 5/2= 7/2. y= 1/2. And then, of course, 1/2+ z= 1/2 gives z= 0. With this added equation x= 7/2, y= 1/2, z= 0 is the unique solution. It is one of the many different solutions the two equations alone gave.

Quote:
2.

a+b+c+d=4
2a-b+c-2d=5

The above involves 4 variables, The above involves 3 variables, since there are four variables do we have to have 4 Simultaneous equations to calculate value of x,y & z?
To calculate a [b]unique solution, yes. With just the two equations, we can add the two equations to eliminate b: 3a+ 2c- d= 9. We can solve for d as d= 3a+ 2c+ 9. Putting that back into the original first equation, a+ b+ c+ 3a+ 3c+ 9= 4a+ b+ 4c+ 9= 4 or 4a+ b+ 4c= -5. That leads to b= -5- 4a-4c. Now, our solution set has "two degrees of freedom" or is "two dimensional". We can choose any numbers we like for both a and c and solve for b and d.

Quote:
say 3a+2b+2c-d=1
5a+3b-3c+3d=3
Okay, those two equations would select a unique solution out of the infinite (two dimensional) set of solutions before.

Quote:
Question 2 :

Can I have both Upper and Lower case Letters in Simultaneous Equations?

Are these valid examples?

Example 1:

x + X =4
x - X =6

Example 2:

x + Y = 1
x - Y = 7
You can but that would be very bad practice! Technically, different symbols, such as x and X, may represent different values but it is very easy to mistake them for one another. As you may have noticed on this board, it is quite common for people, especially those just beginning algebra, to accidentally write "X" when they mean "x".

Quote:
Question 3:

Since there are 26 Upper case characters A.... Z and 26 Lower case characters a ... z which gives us a count of Total 52 variables, can we construct Simultaneous equations as an example to compute all values of the 52 variables.
Why restrict yourself to the Latin Alphabet? As long as you are clear and consistent, you can use any symbols to represent unknown numbers. If, for example, you call your unknowns , , etc., which is very common, you can have any finite number of variables in any finite number of equations.

About 20 years ago, the United States did a project to "justify" township lines on geodesic maps. If you have ever looked at a detailed geodesic map, you may have noticed that lines marking township boundaries often did not quite "match up" due to slight surveying errors (You can imagine the problems surveying land at the crest of the Rocky mountains!). Rather than go back and resurvey everything, the Department of the Interior solved several thousand equations in several thousand unknowns, using statistical methods to match everything up with minimum error.

Quote:
So do we have to write 52 Simultaneous equations in this case ?
If you want a unique solution, yes, with 52 unknown numbers, you would need 52 independent equations. (Obviously "x+ 2y= 3" and "x+ 2y= 3" would not count as two equations. Similarly, "x+ 2y= 3" and "2x+ 4y= 6" would not count as independent equations.)

Quote:
Thanks & Regards,
Prashant S Akerkar
[/quote:1j2xmt2o][/quote:1j2xmt2o]
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August 26th, 2011, 10:28 AM   #3
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Re: Simultaneous Equations

Thank you.

Thanks & Regards,
Prashant S Akerkar
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August 29th, 2011, 02:17 AM   #4
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Re: Simultaneous Equations

Thank you.

Can we come to a conclusion and note down the laws of Simultaneous Equations?

The Number of Simultaneous Equations is greater than or equal to the Number of computing finite unknown variables.

OR

The Number of Simultaneous Equations is directly proportional to the Number of computing finite unknown variables.

As a example, if there are three variables to be computed in simultaneous equation/s, then we have to write three or more Simultaneous Equations.

Would anyone like to make amendments in the Simultaneous Equations Laws?

Thanks & Regards,
Prashant S Akerkar
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August 29th, 2011, 02:31 AM   #5
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Re: Simultaneous Equations

Quote:
Originally Posted by prashantakerkar

Can we come to a conclusion and note down the laws of Simultaneous Equations?
Yes. The number of simultaneous equations must be greater than, equal to, or less than the number of variables. Of course you might not be able to get a unique solution.
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August 29th, 2011, 04:16 AM   #6
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Re: Simultaneous Equations

Read this section of the wiki article
http://en.wikipedia.org/wiki/System_of_ ... l_behavior

Then read the whole article.

Then read a book on Linear Algebra.
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