NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Linear Equations in Two Variables

An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables

Important points to Note

Linear Equations in Two Variables
S. noPoints
1A linear equation in two variable has infinite solutions
2The graph of every linear equation in two variable is a straight line
3x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis
4The graph x = a is a line parallel to y -axis.
5The graph y = b is a line parallel to x -axis
6An equation of the type y = mx represents a line passing through the origin.
7Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph
S. No Type of Equation Mathematical Representation Solutions
S. noType of equationMathematical

representation

Solutions
1Linear equation in one Variable

a and b are real

number

One solution
2Linear equation in two Variable

a, b and c are real

number

Infinite solution possible
3Linear equation in three Variable

a, b, c and d are real

number

Infinite solution possible

Simultaneous pair of Linear equation:

A pair of Linear equation in two variables

Graphically it is represented by two straight lines on Cartesian plane.

Graphically It is Represented by Two Straight Lines on Cartesian Plane
Simultaneous pair of Linear equationConditionGraphical representationAlgebraic interpretation

Example

Intersecting lines. The intersecting point coordinate is the only solution
Intersecting Lines
One unique solution only

Example

Coincident lines. The any coordinate on the line is the solution.
Coincident Lines
Infinite solution.

Example

Parallel Lines
Parallel Lines
No solution

The graphical solution can be obtained by drawing the lines on the Cartesian plane.

Algebraic Solution of system of Linear equation

Algebraic Solution of System of Linear Equation
S. noType of methodWorking of method
1Method of elimination by substitution(1) Suppose the equation are

(2) Find the value of variable of either x or y in other variable term in first equation

(3) Substitute the value of that variable in second equation

(4) Now this is a linear equation in one variable. Find the value of the variable

(5) Substitute this value in first equation and get the second variable

2Method of elimination by equating the coefficients(1) Suppose the equation are

(2) Find the LCM of and . Let it k.

(3) Multiple the first equation by the value

(4) Multiple the first equation by the value

(5) Subtract the equation obtained. This way one variable will be eliminated and we can solve to get the value of variable y

(6) Substitute this value in first equation and get the second variable

3Cross Multiplication method(1) Suppose the equation are

(2) This can be written as

(3) This can be written as

(4) Value of x and y can be find using the

x first and last expression

y second and last expression

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