This document discusses linear equations. It defines linear equations as algebraic equations with terms that are constants or the product of constants and variables. Linear equations can have one or more variables. The document describes variables, constants, and examples of linear equations with one and two variables. It explains how to graph and solve systems of linear equations using graphical and algebraic methods like elimination and cross multiplication. Graphical methods involve plotting the lines defined by each equation and finding their point(s) of intersection. Algebraic methods eliminate variables to solve for the remaining ones.

2. LINEAR EQUATION
A Linear Equation is an algebric equation in
which terms are a constants or the product
of a constants and variables.
Linear Equations can have one or more
variables.

3. VARIABLES & CONSTANTS
• VARIABLES are the unknown part of a Linear
Equation , they are represented with
alphabets . Like->x, y, z, a, b, c, etc.
• Constants are the fixed parts of Linear
Equations . The constants may
be numbers, parameters, or even non-
linear functions of parameters, and the
distinction between variables and parameters
may depend on the problem .

5. LINEAR EQUATIONS IN ONE
VARIABLE
The equations which can be written in the form ->
ax + b = 0 , where a ≠ 0
These type equations are called Linear Equations In One
Variable .
Examples ->
1) 5x + 9 = 0
2) 39a – 5 = 0
3) 345u = -234
4) 5z = 0
ETC.

6. LINEAR EQUATIONS IN TWO
VARIABLES
The equations
which can be
written in the
form->
ax + by – c = 0 ,
where a & b
both can never
be 0
These type
equations are
called Linear
Equations In
Two Variable .
Examples ->
•47x + 7y = 9
•73a – 61b = – 13
•44u + 10v – 155 = 0
•30p + 100 q = 0

7. GRAPHS OF LINEAR EQUATIONS IN ONE &
TWO VARIABLES
ONE VARIABLETWO VARIABLE

8. PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
Each linear equation in two variables defined a straight line.
To solve a system of two linear equations in two
variables, we graph both equations in the same
coordinate system. The coordinates of any points that
graphs have in common are solutions to the system,
since they satisfy both equations. The general form of a
pair of linear equations in two variables x and y as
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
where a1, a2, b1, b2, c1, c2 are all real numbers and
a1
2 + b1
2 ≠ 0 and a2
2 + b2
2 ≠ 0.

9. METHODS FOR SOLVING
PAIR OF LINEAR EQUATIONS IN
TWO VARIABLES
There are two methods for solving PAIR OF
LINEAR EQUATIONS IN TWO VARIABLES

(1) GRAPHICAL Method
(2) ALGEBRAIC Method

10. GRAPHICAL METHOD FOR SOLVING
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
When a pair of linear equations is plotted, two lines are defined. Now,
there are two lines in a plane can intersect each other, be parallel to each
other, or coincide with each other. The points where the two lines intersect
are called the solutions of the pair of linear equations.
Condition 1: Intersecting Lines
If a1a2a1a2 ≠ b1b2b1b2 , then the pair of linear equations a1x + b1y + c1 =
0, a2x + b2y + c2 = 0 has a unique solution.
Condition 2: Coincident Lines
If a1a2a1a2 = b1b2b1b2 = c1c2c1c2 , then the pair of linear equations a1x +
b1y + c1 = 0, a2x + b2y + c2 = 0 has infinite solutions.
Condition 3: Parallel Lines
If a1a2a1a2 = b1b2b1b2 ≠ c1c2c1c2 , then a pair of linear equations a1x +
b1y + c1 = 0, a2x + b2y + c2 = 0 has no solution.
A pair of linear equations which has no solution is said to be an Inconsistent
pair of linear equations. A pair of linear equations, which has a unique or
infinite solutions are said to be a Consistent pair of linear equations.

11. GRAPHS OF ALL THREE CONDITIONS
Intersecting
Lines
Coincident Lines

12. ALGEBRAIC METHOD FOR SOLVING
PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
There are three Algebraic Methods for solving PAIR
OF LINEAR EQUATIONS IN TWO VARIABLES 
1. Elimination by Substitution Method
2. Elimination by Equating Coefficient Method
3. Cross Multiplication Method

13. (1) Elimination by Substitution
Method
Steps 
1. The first step for solving a pair of linear equations by the
substitution method is to solve one equation for either of
the variables.
2. Choosing any equation & any variable for the first step does
not affect the solution for the pair of equations .
3. In the next step, we’ll put the resultant value of the chosen
variable obtained in the chosen equation in another
equation and solve for the other variable.
4. In the last step, we can substitute the value obtained of one
variable in any one equation to find the value of the other
variable.

14. (2) Elimination by Equating Coefficient
Method
Steps 
1. Equate the non-zero constants of any variable by
multiplying the constants of a same variable in both
equations with other equation, so that the resultant
constants of one variable in both equations become
equal.
2. Subtract one equation from another, to eliminate a
variable and find the value of that variable
3. Solve for the remaining variable by putting the value
of one solved variable .

15. (3) Cross Multiplication Method
1)) Let’s consider the general form of a pair of linear equations
a1x + b1y + c1 = 0 , and a2x + b2y + c2 = 0.
2)) To solve the pair of equations for x and y using cross-
multiplication, we’ll arrange the variables x and y and their
coefficients a1, a2, b1 and b2, and the constants c1 and c2 as
shown below
3)) Now simplifying the above situation, and putting the values of x
with 1 & y with 1 to find the value of x & y
x / (b1*c2-b2*c1) = y / (c1*a2- c2*a1) = 1 / (a1*b2-a2*b1)

16. (3) Cross Multiplication Method
Continued
These are the
steps as like
shown in the
picture.
Description
on
corresponding
before page

17. EXAMPLES
Elimination by Substitution
Method

18. Elimination by Equating Coefficient
Method

19. CROSS MULTIPLICATION METHOD

20. Prepared By 
BHAVYAM ARORA
Roll No.  37
Class 10 ‘A’
Submitted To 
Mr. SATYAVEER Sir