Some Examples Of Linear Equation In Two Variables

Factorisation Of a Linear equation In Two Variable:

(NOTE: Eliminate one unknown variable either x or y from both the equations.)

(a) General Equation Problem

a₁x + b₁y + c₁ = 0.................(i)
a₂x + b₂y + c₂ = 0................(ii)  	
Now, eliminate y from both the equations (i) and (ii)
Multiply eq-(i) by b₂ and eq-(ii) by b₁
 
				
a₁b₂x + b₁b₂y + c₁b₂ = 0
a₂b₁x + b₂b₁y + c₂b₁ = 0
 (-)    (-)     (-)
(a₁b₂-a₂b₁)x + (c₁b₂-c₂b₁) = 0

So x is equal to:
x=-(c₁b₂-c₂b₁)/(a₁b₂-a₂b₁)..................(iii)

Now, eliminate x from both the equations (i) and (ii)
Multiply eq-(i) by a₂ and eq-(ii) by a₁

a₁a₂x + b₁a₂y + c₁a₂ = 0
a₂a₁x + b₂a₁y + c₂a₁ = 0
 (-)    (-)     (-)
(b₁a₂-b₂a₁)y + (c₁a₂-c₂a₁) = 0

So y is equal to:
y=-(c₁a₂-c₂a₁)/(b₁a₂-b₂a₁)....................(iv)

The equation (iii) and (iv) represent 
			the values of unknown variables x and y.

(b) Numeric Problem

3x + 2y + 6 = 0...............(i)
5x + 3y + 7 = 0................(ii)
Now, eliminate y from both the equations (i) and (ii)
Multiply eq-(i) by 3 and eq-(ii) by 2

 9x + 6y + 18 = 0
10x + 6y + 14 = 0
(-)  (-)   (-)
-x  +       4 = 0

So x is equal to:
x=4

Now, eliminate x from both the equations (i) and (ii)
Multiply eq-(i) by 5 and eq-(ii) by 3

15x + 10y + 30 = 0
15x +  9y + 21 = 0
(-)   (-)   (-)
        y +  9 = 0

So y is equal to:
y=-9

Addition Of Two Linear Equation In Two Variable:

(NOTE: Be aware about sign)

Ex:1

 5x +  9y +  6 = 0
 5x +  8y +  7 = 0
10x + 17y + 14 = 0

Ex:2

 6x + 11y -  6 = 0
11x +  8y +  7 = 0
17x + 19y +  1 = 0

Subtraction Of Two Linear Equation In Two Variable:

(NOTE: Be aware about sign)

Ex:1

 5x +  9y +  6 = 0
 7x +  8y -  7 = 0
(-)   (-)   (+)
-2x +   y + 13 = 0

Ex:2

 6x + 11y -  6 = 0
11x +  8y +  7 = 0
(-)   (-)   (-)
-5x +  3y - 13 = 0