Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system simultaneously. There are several methods to solve such systems, and the choice of method depends on the specific characteristics of the system. Here are three common methods for solving systems of linear equations:

   1. Substitution Method:-

      Solve one of the equations for one variable in terms of the other variable. Let's say you have x = f(y). Substitute this expression into the other equations in the system, replacing the variable with the expression found in the previous step.This will create a new equation with a single variable, which you can solve for that variable.After finding the value of one variable, you can substitute it back into any of the original equations to find the other variable.Continue this process until you have values for all the variables.

   2. Elimination Method (also known as the addition-subtraction method):-

  • Align the equations so that the variables are vertically aligned.
  • Multiply one or both equations by constants to create a situation where the coefficients of one of the variables in one equation will cancel out when added or subtracted to the coefficients of the same variable in the other equation.
  • Add or subtract the equations to eliminate one of the variables, resulting in a new equation with a single variable.
  • Solve the new equation for one variable.
  • Substitute this value into one of the original equations to find the other variable.

    3. Matrix Method (Gaussian Elimination):-

       Write the system of linear equations in matrix form, often called an augmented matrix.Use row operations to manipulate the matrix into a triangular form or row-echelon form. This is done by adding or subtracting rows to eliminate variables.Once you have the matrix in triangular form, you can use back-substitution to find the values of the variables.

Here's an example of a system of linear equations and how to solve it using the elimination method:-

System of Equations:-

          2x + 3y = 12

          4x - 2y = 10

Solution using Elimination Method:-

  1. Multiply the first equation by 2 to make the coefficients of x in both equations cancel each other when summed:-     

          4x + 6y = 24

          4x - 2y = 10

  2. Subtract the second equation from the first equation to eliminate x:-

          (4x + 6y) - (4x - 2y) = 24 - 10

           Or, 8y = 14

  3. Solve for y:-          

            y = 14 / 8

            y = 7 / 4

  4. Substitute the value of y into one of the original equations to solve for x:- 

         2x + 3(7/4) = 12

   Or,  2x + 21/4 = 12

   Or,  2x = 12 - 21/4

   Or,  2x = 48/4 - 21/4

   Or,  2x = 27/4

   Or,  x = (27/4) / 2

   Or,  x = 27/8         

So, the solution to the system of equations is x = 27/8 and y = 7/4. (Ans.)