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linear equation in 2 variables

Tutormate > CBSE Syllabus-Class 10th Maths > linear equation in 2 variables


Linear equation in two variables: It is the equation with two variables and with highest

power of  variable or Degree .

General form:

ax2 +bx+c =0 and b 0
e.g .   2x+3y= 4   , 2a +3b =9

 Solution of Linear equation in two variables: The values of  and  which satisfy the given

equation is called solution of the equation in two variables.

e.g.  2x +3y+4=0 has solution x=1 and y =-2.

X= -5 , Y= 2 is also solution of the equation.

  • The linear equation can have infinite solutions.
  • The graph of linear equation is a straight line.
  • The solution of a linear equation in variables is a point on the line graph.
  • Any point on line is the solution of linear equation representing line.

Pair of linear equations in 2 variables:

It contains two linear equations in same two variables

In general it is written as   a1x+b1y+c1 =0  ,  a2 x+b2y+c2 =0
with a1,b1,a2,b2  are real and a12 +b120

Geometric meaning of Pair of linear equations in  variables:

Given pair of equations: a1x +b1y +c1 =0
 a2x +b2y +c2 =0

Case 1:

When two lines intersect at a single point.

  • There is unique solution.
  • Equations are called consistent.
°  a1a2   b1b2 

Case 2:

When two lines coincide with each other

  • There are infinitely many solution
  • Equations are called consistent
a1a2 =b1b2 =c1c2

Case 3:

When two lines are parallel.

  • No solution exist
  • Equations are called inconsistent
a1a2 =b1b2 c1c2

Algebraic method to solve pair of linear equations in  variables:

Substitution method:

  • Find value of in terms of using any equation:
    e.g.  y =c1a1xb1
  • Substitute this

     equation i.e. in

    a2x +b2y+c2=0
  • Find the value of  x(or y)
  • Put value x(or y) in eq in step 1.

Elimination Method:

  • Make coefficients of  x (or y) equal by multiplying with suitable non zero real number.
  • Add or subtract both equations.
  • Get the value of y( or x).
  • Substitute value of y ( or x) in suitable equation and get value of other variable.

Cross multiplication method:

Arrange the coefficients as shown:

And we can write

xb1c2b2c1 = yc1ac2a1 =1a1b2a2b1

Equations reducible to a pair linear equation in two variables:

The equations    a1x +b1y =c1  , a2x +b2y =c2
Substitute   1x = p  ,1y=q
And eq.s reduce to    a1p+b1q+c1 =0                                          a2p+b2q+c2 =0

This is pair of linear equations in variables  and .

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