Simultaneous Processing

In mathematics, the theory of linear systems is the branch of linear algebra, a subject which is fundamental to modern mathematics. Computational algorithms for finding the solutions and that they are an important part of numerical linear algebra, and such methods play a prominent role in engineering. A system of the  non-linear equations can often to be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system. Let us see about the problems in simultaneous processing .

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Simultaneous processing problems 1:

Given are two linear equations, let they are x and y
Steps:

• In the first equation, compare the y in terms of x
• After that finishing the first step replace with the value of y in the second equation .
• Then solve the  x by means of the second equation and getting the  in the  before step.
• Substitution the x value for getting in the earlier step in the first equation and find y.

And the below discussed problems are the simultaneous processing problems:

• Problem:

x - y = 14
2x - y = 28
Solution:
x-y=14................(1)
2x-y=28..............(2)
Step 1:
y=x-14.............(3)
Step 2:
substitute 3 in 2
2x-(x-14)=28
2x-x+14=28
x+14=28
x=28-14
x=14
Step 3:
put x=14 in 1
x-y=14
14-y=14
y=0
Therefore the values of variable is
x=14
y=0

• Problem:

2x+3y= -4 ------------(1)
y=x-3
Solution:
Plug y= x-3 in equation 1
2x+3y= -4
2x+3(x-3)= -4
2x+3x-9= -4
5x-9= -4
5x=-4+9
5x=5
x=5/5
x=1
Plug in x=1 in y=x-3
y=x-3
y=1-3
y= -2

• Problem:

2x+3y=6
x+y=5
Solve for x and y
Since it is the substitution processing  method find any one variable from any of the equation and plug in the other equation.
Here the second equation can be in simplified the  form so find x from that
x+y=5
x=5-y
Now plug in the value of x in first equation
So, 2(5-y) +3y=6
10-2y+3y=6
10+y=6
y=6-10
y=-4
Now plug in the value of x in second equation
So, x+y=5
x-4=5
x=5+4
x=9
So the value of x is 9 and the value of y is -4.

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Simultaneous processing problems 2:

• Problem :

3x + 4y + 11 = 0; 5x + 6y = -7
3x + 4y + 11 = 0----------------------(1)
5x + 6y = -7-----------------------(2)
Solution:
3x + 4y + 11 = 0
3x + 4y = -11
3x + = -11 -4y
x =( -11 -4y)/3------------------------------(3)
Substituting the value of x in equation number 2
Therefore, 5x + 6y = -7
5(-11 -4y/3) + 6y = -7
-55 - 20y/3 + 6y = -7 ( multiplying both the sides by LCM 3)
-55 -20y + 18y = -21
-2y = -21 + 55
-2y = 34
y = 34/-2

y = -17
Now substituting y= 17 in equation number 3
x = -11 -4y/3
x = -11 -4(-17)/3
x = -11+ 68 / 3
x = 57/3
x = 19
Therefore, x = 19
y = -17