This document discusses linear equations in two variables. It defines a linear equation as an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides examples of linear equations and discusses their solutions. It describes three algebraic methods for solving a pair of linear equations: elimination method, substitution method, and cross-multiplication method. It provides examples demonstrating how to use each method to find the solutions for a system of two linear equations.

1. SUBMITTED BY:-
BHASKAR GUPTA
CLASS:- 10th A
In the guidance
of:-
MEERA MEENA

2. Linear Equations
Definition of aLinearEquation
• Alinear equation in two variable xis an
equation that canbe written inthe form
ax+ by+c=0, where a ,b and care real
numbers and a and b is not equal to0.
For example-: 3x+4y+1=0
.

3. Let ax + by + c = 0, where a ,b & c are real numbers
such that a and b ≠ O. Then, any pair of values of x
and y which satisfies the equation ax +by +c=O,is
called asolution of it.
APair of Linear Equation In Two
Variables canbesolved –:
i. ByGraphical Method
ii. ByAlgebraic Method
SOLUTIONS OF A LINEAR EQUATION

4. GRAPHICAL SOLUTIONS OF A LINEAR EQUATION
Let usconsider the following system of linear
equations in two variable
I. 2x-y=-1
II. 3x+2y=9
Now solution of eachequation are tobe taken by-
i. Firstly expressone variable in terms ofother.
ii. Now assignany value to one variable andthen
determine the value of othervariable.
Then plot the equations and determine thesolutions
Of system of linear equation in twovariables

5. For Eq.1
2x - y=-1
y =2x +1
Put x=0
y =2(0) +1 y =1
Put x= 2
y =2(2) + 1 y =5
For Eq.2
3x +2y = 9
y =9 - 3x/2
Put x=3
y =9 - 3(3) /2 y =0
Put x=-1
y =9 - 3(-1) /2 y =6
X 0 2
y 1 5
X 3 -1
y 0 6

6. 2 3 4-4 -3 -2 -1 1
-1
-2
-3
-4
-5
-6
( 2,5)
( -1,6)
( 3,0)
5
4
3
2
( 0,1)1
X=1
Y=3
Isthe
solution
for system
of
equations

7. TYPES OF SOLUTIONS of system of equations
UNIQUE
SOLUTI
ON
consistent
a1/a2 ≠ b1/b2
Linesintersecting at
apoint
INFINITE
SOLUTIO
NS
consistent
a1/a2 =b1/b2 =c1/c2
Coincident Lines
NO
SOLUTI
ON
Non
consisten
t
a1/a2 =b1/b2 ≠ c1/c2 Parallel Lines

8. TYPESOFMETHOD:-
Tosolveapairoflinearequationintwo
variablebyALGEABRICMETHOD.
I. Elimination Method
II. Substitution Method
III.Cross-Multiplication Method

9. • Make one variable equal in both theequations
and then either add or subtract the equations
to eliminate thevariable.
• Theresulting equation in one variable is
solved and then by substituting this valueof
the variable in either of the given equation,
the value of the other variable is also
obtained.
ELIMINATIONMETHOD

10. Q.Solveusing the method of Elimination
I. 2x +y = 8
II. 2(x +6y = 15)
For making the coefficients of xin eq.(1) and eq.(2)equal,
we multiply eq.(2) by 2 andget
-(3)
-(4)
2x +y = 8
2x +12y = 30
Subtracting Eq.(3) from eq.(4), we have
11y =22
y =2
Put this value of y ineq.(1)
2x +2 = 8
2x =6 x=3

11. •Find the value of one variable in the terms of
other variable.
Substitute it in other equation and we will get
value of one of thevariable.
Then put the value of variable in any one of the
equation and the value of other variable is also
obtained.
SUBSTITUTIONMETHOD

12. Q.Solveusing the method of Substitution
I. x+2y = -1
II. 2x - 3y =12
y =-2
From Eq.(1), we have
x=-1 – 2y - (3)
Substituting this value of xin eq.(2), wehave
2(-1 – 2y) – 3y =12
- 2 - 4y – 3y =12
- 7y =14
Put this value of y in Eq.(3), wehave
x=-1 -2(-2)
x=-1 + 4 x=3

13. CROSS-MULTIPLICATIONMETHOD

14. Q.Solveusing the method of cross-Multiplication
5x +3y = 35
2x +4y = 28
5x +3y - 35 = 0
2x +4y - 28 = 0
a1 =5
a2 =2
b1 =3
b2 =4
c1=-35
c2=-28
x = y = 1
(3)(-28)-(4)(-35) (-35)(2)-(-28)(5) (5)(4)-(2)(3)
-84 +140 -70 +140 20 – 6
56/14 70/14 14
x=4and y =5