In elimination method,eliminate one variable, to get a linear equation in one variable.

Step 1 : First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.

. Step 2 : Then add or subtract one equation from the other so that one variable gets eliminated. If you get an equation in one variable, go to Step 3.

If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.

If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent.

Step 3 : Solve the equation in one variable (x or y) so obtained to get its value.

Step 4 : Substitute this value of x (or y) in either of the original equations to get the value of the other variable.


Example: The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save ₹ 2000 per month, find their monthly incomes.

Solution : Let us denote the incomes of the two person by ₹ 9x and ₹ 7x and their expenditures by ₹ 4y and ₹ 3y respectively. Then the equations formed in the situation is given by :

	            ${9x – 4y =  2000}$                                                 (1)
		and  ${7x – 3y =  2000 }$                                               (2)
		
Step 1 : Multiply Equation (1) by 3 and Equation (2) by 4 to make the coefficients of ${y}$ equal.
        Then we get the equations:
		${27x – 12y =  6000}$                                                        (3)
		${28x – 12y =  8000}$                                                        (4)
		
Step 2 : Subtract Equation (3) from Equation (4) to eliminate y, because the coefficients of y are the same. 
       So, we get
	   ${(28x – 27x) – (12y – 12y)}$ =  8000 – 6000 
	   i.e.,  ${x}$ =  2000
	   
Step 3 : Substituting this value of x in (1), we get
		${9(2000) – 4y =  2000}$ 
		i.e.,${ y =  4000}$
		

So, the solution of the equations is ${x}$ = 2000, ${y}$ = 4000. Therefore, the monthly incomes of the persons are ₹ 18,000 and ₹ 14,000, respectively.

Verification : 18000 : 14000 = 9 : 7. Also, the ratio of their expenditures = 18000 – 2000 : 14000 – 2000 = 16000 : 12000 = 4 : 3

Example: Use elimination method to find all possible solutions of the following pair of linear equations : 

Solution :
 
	${2x + 3y =  8}$                                                             (1)
	${4x + 6y =  7 }$                                                            (2) 

Step  1  :  Multiply  Equation  (1)  by  2  and  Equation  (2)  by  1  to  make  the coefficients of ${x}$ equal.
 Then we get the equations as :
	${4x + 6y =  16}$                                                            (3)
	${4x + 6y =  7}$                                                             (4)

Step 2 : Subtracting Equation (4) from Equation (3),        
	${(4x – 4x) + (6y – 6y)}$ =  16 – 7
	i.e.,0 =  9, which is a false statement. Therefore, the pair of equations has no solution.

${(10x + y) + (10y + x) =  66}$ i.e., ${11(x + y) =  66}$

i.e., ${x + y =  6 }$                   (1)

We are also given that the digits differ by 2, therefore,

either    ${x – y =  2}$       (2) 

or  ${y – x =  2}$        (3) 

If ${x – y = 2}$, then solving (1) and (2) by elimination, we get ${x=4}$ and ${y=2}$.

In this case, we get the number 42.

If ${y – x = 2}$, then solving (1) and (3) by elimination, we get ${x=2}$  and  ${y=4}$.
 
In this case, we get the number 24.

Answer:Thus, there are two such numbers 42 and 24.