Algebraic Method of solving Linear Equations in Two Variables

In the earlier we covered the first method of solving Linear Equations in Two Variables, i.e. the Graphical Method.

Now let move on to the second method of solving Linear Equations in Two Variables, i.e the Algebraic Method.


In a case when the point of intersection of the lines represented by two given equations has coordinates as rational numbers, graphical method is not convenient. This is when we use the a Algebraic method which is a precise method to obtain an accurate answer.

The most commonly used Algebraic Methods of solving Linear Equations in two variables are as follows:

• Method of Elimination by Substitution.
• Method of Elimination by Equating the coefficients.
• Method of Cross Multiplication.

Method of Elimination by Substitution: Elimination by substitution is the method where we express one of the variables in terms of the other variable from either of the two equations and then this expression id put in the other equation to obtain an equation in one variable.

Method of Elimination by Equating the coefficients: In this method, we eliminate one of the two variables to obtain as equation in one variable which can easily be solved.Putting the value of this variable in any one of the given equations, the value of the other variable can be obtained.

Graphical Method of solving Linear Equations in Two Variables

In the earlier posts we discussed about what is a Linear Equation. Further we also understood the meaning of Linear equations in Two variables and listed the various methods to solve the same.

Today let us analyze the first method that is the Graphical Method of Solving Linear Equation in two variables:

Graphical Method: The Procedure of solving a system of Linear Equations in two variables by drawing their graphs is known as Graphical Method.

The following Algorithm can be used to solve a system of Linear Equations in two variables by graphing method:-

Step 1:- Obtain the given system of Simultaneous Linear Equation in x and y.
Let the system of Linear Equation be

A₁ x +b₁ y ₌ c_{1 .....(i)}

A₂ x +b₂ y ₌ c₂ ...(ii)

Step 2:- Draw the graphs of the equations (i) and (ii) in step 1.
Let the lines l₁ and l₂ represent the graphs of (i) and (ii) respectively.

Step 3:- If the lines l₁ and l₂ intersect at a point and (a,b) are the coordinates of this point, then the given system has a unique solution given by x = a,y = b. Otherwise go to step 4.

Step 4:- If the lines l₁ and l₂ are coincident, then the system id consistent and has Infinitely many solutions. In this case, every solution of one of the equations is a solution of the system. Otherwise got to step 5.

Step 5:- If the lines l₁ and l₂ are parallel, then the given system of equations id in consistent i.e.its has no solution.