Pair of Linear Equations in Two Variables refer to those equations that can be expressed in the form of ax+by+c=0, where a and b are not zero, and a, b and c are real numbers. The graph of a linear equation in two variables is a straight line that is parallel to either the x-axis or y-axis. The pair of values for x and y are the solutions that make each side of the linear equation equal when the value is put in the equation. 

NCERT Solutions Class 10 Maths Pair of Linear Equations in Two Variables serve as an introduction to the CBSE students to linear equations and the solutions to them. The chapter summarizes the graphical and algebraic solutions and Equations Reducible to a Pair of Linear Equations in Two Variables. 

Let's have a look at the topics below to obtain a better understanding of these topics.

The graphical or geometrical solution to a pair of linear equations in two variables is two straight lines that are considered together. We know that for two lines in the same plane, there is a possibility that they will at one point:

• Intersect each other
• Will be coincident
• Will not intersect if the lines are parallel

Here you can see all the different possibilities:

The solution to a pair of linear equations is found by putting the value of one variable at a time to find the value of the other variable at that point.

There are three main methods to find the solution of a pair of linear equations that are:

1. Cross-multiplication Method

         Suppose the equation of a pair of linear equations:

          a1x + b1y + c1 = 0 , and a2x + b2y + c2 = 0.

          In this method, the equations are solved by cross-multiplying the coefficients of x and y as shown in the image.

2. Elimination Method

    In this method, the first step is to multiply the equations to make the coefficient equal for one variable, either x or y. The                  constants that are multiplied should not be zero. 

    Next, from the resultant equations, one equation is subtracted from the other such that the variables with the same constant in      both equations are eliminated. Then, the equation can be solved for the remaining variable. 

    In the last step, the found value can be substituted in any of the original equations to find the value of the other variables.

3. Substitution Method

    In the substitution method, the first step is to solve one equation for one of the variables, x or y. Next, the new value found of          the variable x or y is substituted in the other equation to solve for the variable. In the end, the value found can be substituted in      any one of the equations to find the value of the second variable.

 

Some of the equations that are not linear can be reduced to linear equations with two variables by making a correct substitution to the equation. Once the equations are reduced to linear form, the equations can be solved using any of the three algebraic methods to find the solution of the variable. For example, equations such as: 

8/x + 9/y = 10

In this equation, we can see that it is not linear, but if we substitute the x and y with suitable variables, we can reduce it to a linear equation. 

Here the 1/x can be changed to a, and the 1/y can be changed to b to form a linear equation. So the equation becomes: 

8a + 9b = 10

It makes it very easy to find the value of a and b and then the value of x and y.

Linear equations are equations that have two variables and are in the form of  ax+by+c=0, with a and b never zero, and are real numbers. The graph of a linear equation is a straight line either parallel to the y-axis and x-axis. The solution to a pair of linear equations refers to the pair of values of each variable that fulfills the two equations. There are two main methods of solving the equations, namely, geometrical and algebraic. 

In a geometrical method, the value of one variable is assumed to find the value of the other variable at that point. NCERT Solutions Class 10 Maths Pair of Linear Equations in Two Variables has a wide range of solved examples and questions that are solved by this method. 

In the algebraic method, there are three ways of finding the solution to the pair of linear equations: elimination method, substitution method, and cross-multiplication method. 

Some of the equations are not linear but can be solved by the methods of solving linear equations by substituting the suitable variable to make the equation linear. The examples of each method can be found in the NCERT solutions class 10 pair of linear equations below. 

Here are the NCERT solutions class 10 Pair of linear equations questions with different methods. 

Example 1

Example 2

Example 3 

Example 4

Example 5

1. What is a linear equation in two variables Class 10? 

A linear equation is an equation that contains two variables, standardly taken as x and y, with real numbers a, b, and c, where a and b are non-zero numbers. 

The standard form of a linear equation is ax + by + c = 0 or ax + by = c.

2. What are the important topics in linear equation in two variables Class 10? 

    The topics covered in the Class 10 Maths Pair of Linear Equations in Two Variables provide a solid base for further topics and            understanding graphical representation. The important topics in the chapter are as follows:

1. Introduction to Pair of Linear Equations in Two Variables 

2. Graphical Method of Solution of a Pair of Linear Equations 

3. Algebraic Methods of Solving a Pair of Linear Equations 

4. Equations Reducible to a Pair of Linear Equations in Two Variables

 

3. What are two examples of equations in two variables?

Linear Equations in Two Variables are equations that are in the form of ax + by + c = 0 or ax + by = c where x and y are two variables and a, b, and c are real numbers. For example,

1. 9x + 6y = 8 is a linear equation in two variables in the form of ax + by = c. 

2. 7x + 3y - 1 = 0 Is in the form ax + by + c = 0.