RS Aggarwal Solutions Class 10 Chapter 3 - Linear Equations in two variables (Ex 3I) Exercise 3.9

A system of equations can be called the Linear Equations in two variables if they have either a unique solution, no solution or infinitely many solutions. A system of Linear Equations may have ‘n’ number of variables. While we solve the Linear Equations with n number of variables, there must be n equations to solve for the values of the variables. The set of solutions that are obtained on solving these Linear Equations is a straight line and Linear Equations in two variables are the algebraic equations which are of the form y = mx + y where m is the slope and y is the y intercept of the coordinates. These are equations of the first order. 

Graphical Method

• Step 1: To solve a system of Linear Equations in two variables, we represent each of the equations in a standard graph.

• Step 2: To represent a  graph of the given equation, we first convert it to the form of y=mx+b by solving the given equation for y.

• Step 3: Then, we substitute the values of x as 0, 1, 2, 3, and so on to find the corresponding values of y, vice-versa.

• Step 4: Identify the point where both the lines meet.

• Step 5: The point of intersection of both the lines is the solution of the given system of Linear Equations in two variables. 

But, in some cases, both lines may not always intersect. Sometimes they may be parallel to each other. In that case, the system of Linear Equations in two variables will have no solution. In some of the other cases, both lines coincide with each other. In those cases, each point on that particular line is a solution of the given system and hence, the given system has infinitely many numbers of solutions. If the system has a solution, then it is said to be a consistent system; otherwise, it is said to be an inconsistent system. We can identify a system of Linear Equations in two variables if the equations that are expressed in the form ax+by+ c = 0, which consist of two variables x and y and the highest degree of the given equation is 1.

Substitution Method

• Step 1: Solve one of the given equations for one variable.

• Step 2: Then, substitute this variable into the other equation to get an equation in terms of a single variable.

• Step 3: Solve this equation for getting the next variable.

• Step 4: Then, substitute it in any of the equations to get the value of another variable.

Elimination Method

• Step 1: Arrange the given equations in the standard form ax+by+c=0 or ax+by=c.

• Step 2: Check all possibilities of adding and subtracting the equations which would result in the cancellation of a variable.

• Step 3: If it does not, multiply one or both the equations by either the coefficient of x or coefficient of y, so that their addition or subtraction would result in the cancellation of any one of the variables.

• Step 4: Solve the resulting equation with one variable.

• Step 5: Substitute it in any given equation to get the value of another variable.

Graphical Representation of Linear Equations in Two Variables

We can represent the Linear Equations in two variables graphically using the following steps:

• Step 1: Linear Equations in two variables can be represented graphically by graphing each equation by converting it to the form y=mx+b by solving the equation for the y variable.

• Step 2: Identify the points at which both of the lines are intersecting.

• Step 3: The point of intersection is the solution of the given system of the Linear Equations in two variables.