If one is in ratio, the other one is zero

Using this formula we can solve simultaneous linear equations which may involve big numbers.In special cases,we can solve these equations by just looking at it.

Example 1:

5x + 4y =6

27x + 20 y =30

Observe the above equation carefully.You can notice that the ratio of coefficients of y is same as that of the constant terms .i.e;coefficients of y is 4 :20  i.e; 1:5 which is same as the constant terms 6:30 i.e; 1:5 .Hence we put x = 0.

Put x =0 in any equation above and calculate mentally the value of  y. Doing the same in the first equation we get y= 6/4.

Example 2:

2 x + 4 y =3

16x+5y=24

Here, coefficients of x is 2:16 i.e; 1:8 which is same as the constant term 3:24 .i.e; 1:8 meaning that the ratio of coefficients of x is same as that of the constant terms .So we put  y = 0 in any equation above.

Soving the above equations we get the value of x = 3/2

Example 3:


In solving simultaneous quadratic equations too, we can make use of this sutra
Solve x + 4y = 10 and  x² + 5xy + 4y² + 4x - 2y = 20 ?
First lets simplify this  x² + 5xy + 4y² + 4x - 2y = 20
 = ( x + y ) ( x + 4y ) + 4x – 2y = 20

 = 10 ( x + y ) + 4x – 2y = 20 ( Since x + 4y = 10 )

 = 10x + 10y + 4x – 2y = 20

 = 14x + 8y = 20

Now comparing these two equations x + 4y = 10  and 14x + 8y=20 ,we see that coefficients of y is same that of the constant term(4 : 8 :: 10 : 20) so we put x=0 to get the value of y.Therefore,value of y = 5/2
            

Note:
This technique works  ONLY when either the co-efficients of x or y equals to that of the constant terms.This can be extended to more general cases with any number of variables

Example :

ax + by + cz = a

bx + cy + az = b

cx + ay + bz = c

which yields x = 1, y = 0, z = 0 since the coefficients of x is same that of the constant terms.

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