NCERT Revision Notes for Chapter 3 Pair of Linear Equations in Two Variables Class 10 Maths

Answer

• Linear equation in two variables: An equation in the form of ax + by + c = 0 where x and y are variables and a, b, c are real numbers (a≠0, b≠0) is called linear equation in two variable.

Example: (i) 3x + 4y + 4 = 0

(ii) 2/3 x + y = 0

• Pair of linear equation in two varibales: Two linear equations in the same two variables are called a pair of linear equations in two variables. The general form of a pair of linear equations is:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

where, a1, b1, c1, a2, b2, c2 are real numbers and none of them are equal to zero.

Example:
(i) 3x + 4y + 6 = 0

x + 2y + 3 = 0

Answer

Let two lines on Cartesian plane, each representing a linear equation,

Eq 1:a ₁ x+b₁ y+c ₁=0

Eq 2: a₂ x+b₂ y+c₂=0

Condition 1: Respective pair of Linear Equations will have one unique solution.

The condition is the ratio of respective coefficients of x should not be equal to that of y. The grapha of both equation will intersect at only one point. The coordinate of that points will represent solution of equation 

Condition 2: Respective pair of Linear Equations will have Infinite Solution. 

The condition is the ratio of respective coefficents of x should be equal to that of y, and that should be equal to the ratio of constant terms. The graph of both equations coincides each other. 

Condition 3: Respective pair of Linear Equations will have No Solution

Note: If Graphs of equations have solution they are known as consistent system otherwise they are Inconsistent. First two cases are consistent system.

Answer

Steps: 

I. Find 2 coordinate points Satisfying Equation 1, say (A, B). Draw a line pasing through both these points. Consider this line 1. This can be done by writting one variable in form of another, putting some value for one variable and finding other. Suppose, we write x in terms of y,and put value for y to get respective x. 

II. Locate two points on Cartesian plane. Values obtained for x will be x-coordinate and values obtained for y will be on y-cooridnate.

III. Repeat step I for Equation 2. 

III.  The two lines will intesect each other at some coridinate point. The x-coordiante ot that point  will represent common solution for x and y-coordinate of that point will represent common solution y of both equations. 

Following Graph represent Solution  of two Equations

Eq1: 3x+y-11=0

Eq2: x-y-1=0

Answer

Steps used in this method to solve a pair of linear equations are given below.

Step I: Find the value of one variable say x (or y) in terms of other variable i e . ., y (or x) from an equation.

Step II: Substitute this value of x (or y) in other equation, then it reduces to a linear equation in one variable i.e., in terms of y (or x) which can

be solved easily.

Step III: Substitute the value of y (or x) obtained in step II in the equation which is used to obtain the value of the other variable in step I.

Note: In this method, we have substitute the value of one variable in terms of other variable to solve the pair of linear equations. So, this method is called substitution method.

Answer

Step I: Firstly, make the coefficient of one variable (x or y) numerically equal by multiplying both equations by some suitable non-zero constant.

Step II: Now, add or subtract both equations, so that one variable is eliminated and remaining equation has one variable.

Step III: Solve the equation in one variable to get the value of this variable (x or y).

Step IV: Substitute this value (x or y) in any one of the given equations to get the value of other variable.

Answer

In order to solve, by cross multiplication, for system of equations Following Method is used to write cross multiplication Method

Rule: 

Multiply the co-efficient according to the arrow heads and subtract the upward product from the downward product. Place the three differences under x, y and 1 respectively forming three fractions; connect them by two signs of equality.

Answer

The Pair of Linear Equations is Used to Solve many General life problems based on:

Ages

Boat and Stream

Digits

Fractions