System of linear equations is an essential mathematical concept that is used to solve problems relating to multiple variables simultaneously. This concept is applicable in various fields like science, engineering, business, and finance. In this article, we will learn about the basics of linear equations and how they can be solved using different methods.

A linear equation is an algebraic expression in which the highest exponent of the variable is 1. It can be written in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. For example, 3x + 2y = 7 is a linear equation.

A system of linear equations consists of two or more linear equations with two or more variables. The goal is to find the unique values of the variables that satisfy all the equations in the system. For example, consider the following system of linear equations:

2x + y = 4
x – y = 2

To solve this system, we can use different methods like substitution, elimination, and matrix method.

In the substitution method, we solve one of the equations for one of the variables and substitute its value in the other equation. For example, in the above system, we can solve the second equation for x as x = y + 2 and substitute in the first equation to get 2(y+2) + y = 4. Simplifying, we get 3y + 4 = 4, which gives y = 0. Substituting y = 0 in x = y + 2, we get x = 2. Hence, the solution of the system is (2,0).

In the elimination method, we multiply one or both equations by a constant to make the coefficients of one of the variables equal and then subtract one equation from the other to eliminate one variable. For example, we can multiply the first equation by 2 to get 4x + 2y = 8 and subtract the second equation to get 3y = 4. Solving for y, we get y = 4/3. Substituting y = 4/3 in the second equation, we get x = 8/3. Hence, the solution of the system is (8/3,4/3).

In the matrix method, we write the coefficients of the variables in a matrix form and then perform row operations to transform the matrix into a row echelon form, from which we can read the solution of the system. For the above system, the augmented matrix is given by:

Performing row operations, we get:

From the row echelon form, we can read the solution as x = 8/3 and y = 4/3, which is the same as obtained by the elimination method.

In conclusion, the system of linear equations is a fundamental concept in mathematics that is used to solve problems involving multiple variables. The methods of solving such systems include substitution, elimination, and matrix method. It is important to note that not all systems have unique solutions, and some may have no solutions or infinitely many solutions depending on the coefficients of the equations. Hence, it is crucial to understand each method and its limitations to solve the system accurately.

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