METHODS OF SOLVING LINEAR EQUATIONS IN TWO VARIABLES

General form of linear equation in two variables :

ax + by + c  =  0

Different methods of solving linear equations :

(i)  Substitution method

(ii)  Elimination method

(iii)  Cross multiplication method

(iv)  Graphical method

Solving Linear Equations Using Substitution method

Step 1 : 

In the given two equations, solve one of the equations either for x or y. 

Step 2 : 

Substitute the result of step 1 into other equation and solve for the second variable. 

Step 3 : 

Using the result of step 2 and step 1, solve for the first variable.  

Examples of solving linear equations using substitution method

Solving Linear Equations Using Elimination Method

Procedure for elimination method :

• Using the elimination method, we can  eliminate any one of the variables by combining both equations. 
• We can do elimination if and only if the coefficients are same. If they are different, then we have to make the coefficients as same and then we may continue elimination.
• If the coefficients along with the signs are same, we may do elimination by subtracting both the equations.
• If the signs are different and coefficients are same, then we may do elimination by adding both the equations.
• By applying the value of variable we get from the previous step in either of the equations, we will get the value of the remaining variable.

Solving linear equations using elimination method

Solving Linear Equations Using Cross Multiplication Method

Let us consider the following system of linear equations. 

a₁x + b₁y + c₁  =  0

a₂x + b₂y + c₂  =  0

We have to write the coefficients of the equations and do cross multiplication as shown below. 

We write the coefficient of y and constant term and two more columns by repeating the coefficients of x and y as follows. 

The result is given by

The solution is

Solving linear equations using cross multiplication method

Solving Linear Equations Using Graphical Method

Let the system of pair of linear equations be

a₁x + b₁y  =  c₁

a₂x + b₂y  =  c₂

We know that the given two lines in a plane only one of the following may happen

(i)  Two lines will intersect at one point

(ii)  Two lines will not intersect

(iii)  Two lines are coincident

Examples of the solving linear equations using graphical method

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