In mathematics, algebraic expressions and polynomials are fundamental concepts in algebra. Also, one might know that the value of an algebraic expression depends on the values of the variables involved. A polynomial in one variable and degree 1 is called a linear polynomial in one variable. When we equate this polynomial to 0, we get the linear equation. Thus, when an equality sign separates two expressions, it is called an equation. Similarly, we can define the linear equations in two variables. Solving these linear equations, we can get the values of variables as a solution. We have different techniques to solve the given linear equations with simple steps.

The purpose of solving a linear equation is to locate the variable’s value that will make the statement, i.e. the equation true. As we know, linear equations can be solved in either of the following methods:

• Graphical Method (involves plotting points and drawing lines)
• Elimination Method (involves the elimination of one variable)
• Substitution Method (involves the substitution of expression derived from one of the equations)
• Cross Multiplication Method (formula)
• Matrix Method (finding the matrix inverse)
• Determinants Method (ratio of determinants of matrices)

Among these methods, the matrix method is quite different, and it can be done by evaluating the inverse matrix after converting the given linear equations to matrix form. However, this method is the most efficient technique for solving linear equations in two or three variables.

Let’s understand the various methods of solving linear equations in two variables in brief. The graphical method is generally used to solve the linear equations in two variables. In this method, we plot the points as x-coordinate and y-coordinate that satisfy the given equation. Then, we need to draw the lines passing through the respective points, and the intersection of the lines corresponding to the given equations will define the solution set. Now, coming to the elimination method, first, we need to equate and eliminate one of the variable’s coefficients. Then, the given equations are solved to get the other equation. In the substitution technique of solving given linear equations, the first step is to isolate the value of one variable from any of the equations. Later, we substitute the isolated variable’s value in another equation and solve it. 

We can also solve linear equations in two variables using a formula, and this is defined in the cross-multiplication technique. Thus, this method of solving linear equations involves the substitution of coefficients of variables in the defined formula that is again in the form of an equation. When it comes to the matrix method of solving linear equations, we generate the 2×2 matrix and evaluate the inverse of 2×2 matrix to get the values of the two variables involved. However, it is easy to find the matrix inverse for 2×2 matrix as this involves simple computation of required components, such as determinant and adjoint of matrices. Finding the adjoint matrix for the 2×2 matrix is quite simple and does not require any calculations.

There exist one more method of solving linear equations, called the determinant method. We must write several matrices from the given equations based on specific rules in this method. Thus, we can solve linear equations using several simple techniques and formulas. Students can learn all these methods in their secondary and senior secondary classes with proper rules or formulas, along with solved examples.