Watch Free Classes
GATE 2026 Exam Date Announced – Complete Schedule, Syllabus, and Key Details » GATE Study Materials » Mechanical Engineering » Solution of Linear Equations
Prepare for GATE & ESE

Solution of Linear Equations

This article will be written on the core mathematical concept of linear equations. This topic is very important in Agricultural engineering. The topic will be properly discussed through analysing linear equations in two variables, linear differential equations, and solutions of linear equations.

Table of Content
A linear equation is normally described as an equation whose highest variable power is always equal to 1. Generally, it is also referred to as an equation of degree one. The standard appearance in which one variable linear equation is often expressed is given as Cx + D = 0. In this standard form, C is regarded as the coefficient, x is the variable whereas D is referred to as the constant. On the other hand, the standard appearance in which a two-variable linear equation is often expressed is given as Cx+Dy=E. Here y, as well as x, are variables, E is the constant whereas C, as well as D, are referred to as coefficients.

Linear equation in two variables

A particular linear equation that involves two variables is generally expressed in the form or given as Cx  + Dy + E = 0. Here, C, D as well as E are all real numbers whereas y as well as x are the two provided variables, each having a degree of 1. Specifically, if we take two equations that are linear and have two variables, then they are referred to as simultaneous linear equations. For instance, 5y + 7x + 8 = 0, 3y + 2x – 7 = 0 are 2 linear equations having two variables. There are several methods through which this two-variable linear equation can be solved. These methods involve the method of substitution, the method of using graphs, the method elimination, the method of determinants as well as the method of cross-multiplication.

Solutions of linear equation

An equation can be described as weighing two balances having equal weights on each side. If we subtract or add a particular number from each side of the provided equation then it is still true. Similarly, if we divide or multiply a particular number from both sides of the provided equation then the equation is again correct. Normally, the variables are brought on one chosen side of the provided equation while the constant is taken on the other side. Next, the value of the unknown variable is computed. This is the process through which solutions of linear equations with one variable can be computed. On the contrary, the solutions of linear equations with two variables can be computed through the method of determinants, the method elimination, and so on. However, to apply these methods two variables are necessary. In the following section, an example in this regard has been provided. Example: Find the solution of 23x – 3 = 20. Solution: Mathematical operations are performed on the RHS (Right-Hand Side) and LHS (Left Hand Side) so the balance remains the same. Hence, at first 3 must be added on both sides of the equations to reduce the LHS of the provided equation to 23x. Thus the new LHS becomes 23x – 3 + 3 = 23x. For this LHS the RHS becomes 20 + 3 = 23. Next, 23 must be divided from LHS and RHS for reducing LHS to x. After doing this we obtain the value of x as 1. This can be considered as an efficient way of solving a one-variable linear equation.

Linear Differential equations

A linear differential equation can be defined as an equation that includes a variable, a derivative, or a differentiation of this particular variable and some associated functions. A linear differential equation’s standard form is given by dy / dx + Py = Q. This standard form also contains the variable y as well as its derivative. In this standard form of a linear differential equation, Q, as well as P, are either functions of x or numeric constants. It can be specifically seen that a linear differential equation is a vital type of differential equation that can be solved via using a particular formula. Some linear differential equation examples have been provided in the following.
  • dy / dx +y = Sinx
  • dy / dx – (2y / x) = e-x . x2

Conclusion

The overall article has been written on the chief topic of solutions of linear equations. In mathematics, the concept of linear equations is very important. Further, in the case of Agricultural Engineering, this topic has great importance. The linear equation is essentially a one-degree equation and the topic of linear equations has been further analysed through the assessment of Linear equations in two variables, solutions of linear equations, and linear differential equations.