linear equation in 2 variables

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03 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:

a x^{2} + b x + c = 0 a n d b \neq 0

e . g . 2 x + 3 y = 4, 2 a + 3 b = 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

I n g e n e r a l i t i s w r i t t e n a s a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0

w i t h a_{1,} b_{1,} a_{2}, b_{2} a r e r e a l a n d {a_{1}}^{2} + {b_{1}}^{2} \neq 0

Geometric meaning of Pair of linear equations in variables:

G i v e n p a i r o f e q u a t i o n s : a_{1} x + b_{1} y + c_{1} = 0

a_{2} x + b_{2} y + c_{2} = 0

Case 1:

When two lines intersect at a single point.

There is unique solution.
Equations are called consistent.

° \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b 2_{}}

Case 2:

When two lines coincide with each other

There are infinitely many solution
Equations are called consistent

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

Case 3:

When two lines are parallel.

No solution exist
Equations are called inconsistent

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

Algebraic method to solve pair of linear equations in variables:

Substitution method:

Find value of in terms of using any equation:
$e . g . y = \frac{c_{1} - a_{1} x}{b_{1}}$

Substitute this
$2^{n d}$
equation i.e. in
$a_{2} x + b_{2} y + c_{2} = 0$

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

\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}

Equations reducible to a pair linear equation in two variables:

T h e e q u a t i o n s \frac{a_{1}}{x} + \frac{b_{1}}{y} = c_{1}, \frac{a_{2}}{x} + \frac{b_{2}}{y} = c_{2}

S u b s t i t u t e \frac{1}{x} = p, \frac{1}{y} = q

A n d e q . s r e d u c e t o a_{1} p + b_{1} q + c_{1} = 0 a_{2} p + b_{2} q + c_{2} = 0

This is pair of linear equations in variables and .

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