A 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.

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What is Linear Equation?

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.

System of Three Linear Equations in Two Variables

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:

1. Independent System: An independent system is considered consistent and has exactly one solution pair. The point where two lines intersect is the only solution on a graph, and the equation falls under an independent system.
   
2. Inconsistent System: Unlike independent systems, the inconsistent system has no solution. Consider two parallel lines in a graph that will never intersect; hence, it is an inconsistent system of linear equations in two variables.
   
3. Dependent System: On the other hand, a dependent system has infinite solutions. In a dependent system, the equations are on the same line; hence, every coordinate pair on the line is a solution to both equations.

Solving Linear Equations in Two Variables

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.

Substitution Method

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:

1. The given equation needs to be simplified by expanding the parenthesis.
2. Now, one of the equations needs to be solved, either x or y.
3. Students need to now substitute the solution into the other equation.
4. Solve the new equation which has been obtained using arithmetic operations.
5. Lastly, solve the equation to find the value of the second variable.

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.

Elimination Method

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:

1. Firstly, the given equation needs to be multiplied by a non-zero constant to make the coefficient numerically equal.
2. Now, add or subtract one equation from the other in such a way that one variable is eliminated.
3. Now get the value of one variable by arithmetic operations.
4. Substitute the obtained value of any of the equations to get the value of the other variable.

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.

Cross Multiplication Method

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:

1. Let us take two linear equations a1x+b1y+c1=0 and a2x+b2y+c2=0.
2. Now write the product of the two by cross multiplication:
   x/b1c2-b2c1 = y/c1a2-c2a1 = 1/a1b2=b2a1
3. Now solve the equations for variable:
   x/b1c2-b2c1 = 1/a1b2=b2a1; y/c1a2-c2a1 = 1/a1b2=b2a1
4. Lastly, find the values of the variable x and y:
   x= b1c2-b2c1/a1b2-b1a2; y= c1a2-c2a1/a1b2-b1a2
   

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