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April 8, 2025A linear equation is a concept in Maths that is introduced in Tamil Nadu Board in Class 9. A linear equation can be defined as an equation that has the highest degree of 1, which means that no variable in a linear equation has an exponent greater than one. Here, the coefficients are often real numbers, and a linear equation gives a straight line when plotted on a graph between two variables.
Students often find the concept of linear equations challenging and hard to master. However, there is no need to worry because this article has explained a system of three linear equations in two variables in-depth. Moreover, this concept will help students build a strong foundation for future Maths classes and a strong understanding of subjects such as Physics and Chemistry.
Before knowing the system of three linear quotations in two variables, students need to understand a linear equation well. In simple words, a linear equation is an equation which is written in the form x+by +c=0. a, b, and c are real values, whereas x and y are variables. One example of a linear equation in two variables would be 2x+y=15.
There are three systems of linear equations in two variables, and they are independent system, inconsistent system, and dependent system. Moreover, there are three systems to solve linear equations in two variables, and they are substitution method, elimination method, and cross multiplication method. Students can find a comprehensive explanation of the systems in this article below.
A system of linear equations means finding a unique way to solve them and finding a numerical value that will satisfy all equations in the system at the same time. A system of linear equations consists of two or more linear equations that are made up of two or more variables, and all the equations are considered simultaneously. For that, we have to consider linear equations by the number of solutions and only then can we determine the system of linear equations in two variables. The three systems of linear equations in two variables are as follows:
There are three ways to solve linear equations in two variables. Students must remember that a and b of the equation are constant, i.e. real numbers. The examples of ways to solve linear equations here are algebraic methods. The three ways to solve linear equations in two variables are given in this article below.
In the substitution method, the value of one variable is substituted from one equation. The steps to solve a linear equation in two variables by substitution method are as follows:
An example of the substitution method is:
Let 7x−15y=2⋯(i) and x+2y=3- (ii)
From equation (ii),
⇒x=3−2y−⋯(iii)
Then, equation (i),
⇒7(3−2y)−15y=2
⇒21−14y−15y=2
⇒−29y=2−21=−19
⇒y=1929
Then, from equation (iii),
⇒x=3−2(1929)
⇒x=87−3829
⇒x=4929
Therefore, the solution of the given equations is x=4929,y=1929.
In the elimination method, to get the equation in one variable, the equation in the other equation is either added or subtracted. The steps to solve a linear equation in two variables by elimination method are as follows:
An example of elimination method is as follows:
Let, x+y=5…..(i) and 2x−3y=4…..(ii)
Equate the co-efficient of variable x, multiply the equation (i) with (ii).
(i)×2⇒2x+2y=10……(iii)
Subtract equations (ii) and (iii),
2x–3y=42x+2y=10(–)_____________–5y=–6
⇒y=6/5
Then, from the equation (i),
⇒x+65=5
⇒x=5−65=25−65
⇒x=19/5
Therefore, the values of x and y are 19/5 and 6/5.
The cross multiplication method is often considered the easiest way to solve linear equations and also the most accurate. The cross multiplication method is only applicable to two variables and the image below will give a better understanding of how the method works.
The steps to solve linear equations by cross multiplication method can be understood properly with an example that is provided below:
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