Solving Simultaneous Equations | Simultaneous Linear equations | Linear equation in two variables.

Linear equations in two variables :
Linear equations in two variables
Definition of simultaneous equations :
" Two equations that are considered at the same time are called simultaneous equations. "
Solution of simultaneous equations :
if each of the equations in a simultaneous equation is satisfied by a common value pair of variables then this pair of variables is called a solution of a simultaneous equation.

 Simultaneous equations examples :

 1) x + y = 100
     x - y = 20
So how many x and y.
Solution :
x + y = 100 -------- I
x - y = 20 --------- II
Adding equation I from equation II , we get
2x = 120
x = 120 / 2
x = 60
Let's put  x = 60 in equation I , we get
60 + y = 100
y = 100 - 60
y = 40
Therefore, values of x and y  is ( 60, 40 ) .
By verification :
The given solution should be kept in the equation. If both the values   are the same then it should be considered as a solution.
Above two simultaneous equations are
x + y = 100 and x - y = 20
put x = 60 and y = 40 in equation I
We get,
x + y = 100
60 + 40 = 100
100 = 100
L.H.S. = R.H.S.
Since both sides are the same, this solution is of that equation.
2) 3x - 2y + 4 = 0 ;   2x - y + 3 = 0
Solve how many x and y.
Solution:
3x - 2y + 4 = 0 ----------I
2x - y + 3 = 0 -----------II
Multiples equation II by 2, we get
4x - 2y + 6 = 0 -----------III
Subtract equation III  from equation I, we get
- x - 2 = 0
- x = 2
x = - 2
Let's put x = -2 in equation I, we get
3 ( - 2 ) - 2 y + 4 = 0
-6 - 2y + 4 = 0
-2y -2 = 0
-2y = 2
- y = 1
y = -1
Therefore, x = -2 and y = -1
By verification :
Above two simultaneous equations are
3x - 2y + 4 = 0    and
2x - y + 3 = 0
Let's put x = -2 and y = -1 in equation I , we get
3 ( - 2 ) -2 ( -1 ) + 4 = 0
-6 + 2 + 4 = 0
-6 + 6 = 0
0 = 0
Therefore, L.H.S. = R.H.S.
Since both sides are the same, this solution is of that equation.
3) 2x + y - 8 = 0 ;   2x - 3y + 8 = 0
Solve how many x and y.
Solution :
2x + y - 8 = 0 ----------I
2x - 3y + 8 = 0 -----------II
Subtract equation II  from equation I, we get
4y - 16 = 0
4y = 16
y = 16 / 4
y = 4
Let's put y = 4 in equation I, we get
2x + 4 - 8 =0
2x - 4 = 0
2x = 4
x = 4 / 2
x = 2
Therefore,  x = 2 and y = 4
By verification :
Abrove two simultaneous equations are
2x + y - 8 = 0    and
2x - 3y + 8 = 0
Let's put x = 2 and y = 4 in equation I , we get,
2x + y - 8 = 0
2 ( 2 ) + 4 - 8 = 0
4 + 4 - 8 = 0
8 - 8 = 0
0 = 0
Therefore, L.H.S. = R.H.S.
Since both sides are the same, this solution is of that equation.


9. Single equation method :

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