LINEAR LINEAR EQUATION IN EQUATION IN

TWO TWO VARIABLESVARIABLES

System of equations or System of equations or simultaneous equations –simultaneous equations –

A pair of linear equations in two A pair of linear equations in two variables is said to form a system variables is said to form a system of simultaneous linear equations.of simultaneous linear equations.

For Example, 2x – 3y + 4 = 0For Example, 2x – 3y + 4 = 0 x + 7y – 1 = 0x + 7y – 1 = 0

Form a system of two linear Form a system of two linear equations in variables x and y.equations in variables x and y.

The general form of a linear The general form of a linear equation in equation in two variables x and y is two variables x and y is

ax + by + c = 0 , a and b is not ax + by + c = 0 , a and b is not equal to zero, whereequal to zero, where

a, b and c being real numbers.a, b and c being real numbers.

A solution of such an equation is a A solution of such an equation is a pair of values, One for x and the pair of values, One for x and the other for y, Which makes two sides other for y, Which makes two sides of the equation equal. of the equation equal.

Every linear equation in two variables Every linear equation in two variables has infinitely many solutions which has infinitely many solutions which can be represented on a certain line.can be represented on a certain line.

GRAPHICAL SOLUTIONS OF GRAPHICAL SOLUTIONS OF A LINEAR EQUATIONA LINEAR EQUATION

Let us consider the following system of Let us consider the following system of two simultaneous linear equations in two simultaneous linear equations in two variable.two variable.

2x – y = -12x – y = -1

3x + 2y = 93x + 2y = 9

Here we assign any value to one of Here we assign any value to one of the two variables and then the two variables and then determine the value of the other determine the value of the other variable from the given equation.variable from the given equation.

For the equationFor the equation

2x –y = -1 ---(1)2x –y = -1 ---(1)

2x +1 = y2x +1 = y

Y = 2x + 1Y = 2x + 1

3x + 2y = 9 --- (2)3x + 2y = 9 --- (2)

2y = 9 – 3x 2y = 9 – 3x

Y = 9-3xY = 9-3x

X 0 2

Y 1 5

X 3 -1

Y 0 6

X X

Y

Y

(2,5)(-1,6)

(0,3)(0,1)

X= 1Y=3

ALGEBRAIC METHODS OF ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS SOLVING SIMULTANEOUS

LINEAR EQUATIONSLINEAR EQUATIONS

The most commonly used algebraic The most commonly used algebraic methods of solving simultaneous methods of solving simultaneous linear equations in two variables arelinear equations in two variables are

• Method of elimination by Method of elimination by substitution.substitution.

• Method of elimination by equating Method of elimination by equating the coefficient.the coefficient.

• Method of Cross - multiplication.Method of Cross - multiplication.

ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION

STEPSSTEPS

Obtain the two equations. Let the Obtain the two equations. Let the equations be equations be

aa11x + bx + b11y + cy + c11 = 0 ----------- (i) = 0 ----------- (i)

aa22x + bx + b22y + cy + c22 = 0 ----------- (ii) = 0 ----------- (ii)

Choose either of the two equations, say (i) Choose either of the two equations, say (i) and find the value of one variable , say ‘y’ and find the value of one variable , say ‘y’ in terms of x.in terms of x.

Substitute the value of y, obtained in the Substitute the value of y, obtained in the previous step in equation (ii) to get an previous step in equation (ii) to get an equation in x.equation in x.

ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION

Solve the equation obtained in the Solve the equation obtained in the previous step to get the value of x.previous step to get the value of x.

Substitute the value of x and get Substitute the value of x and get the value of y.the value of y.

Let us take an exampleLet us take an example

x + 2y = -1 ------------------ (i)x + 2y = -1 ------------------ (i)

2x – 3y = 12 -----------------(ii)2x – 3y = 12 -----------------(ii)

SUBSTITUTION METHODSUBSTITUTION METHOD

x + 2y = -1x + 2y = -1

x = -2y -1 ------- (iii)x = -2y -1 ------- (iii)

Substituting the value of x in Substituting the value of x in equation (ii), we getequation (ii), we get

2x – 3y = 122x – 3y = 12

2 ( -2y – 1) – 3y = 122 ( -2y – 1) – 3y = 12

- 4y – 2 – 3y = 12- 4y – 2 – 3y = 12- 7y = 14 , y = -2 , - 7y = 14 , y = -2 ,

SUBSTITUTIONSUBSTITUTION

Putting the value of y in eq. (iii), we Putting the value of y in eq. (iii), we getget

x = - 2y -1x = - 2y -1

x = - 2 x (-2) – 1 x = - 2 x (-2) – 1

= 4 – 1 = 4 – 1

= 3= 3

Hence the solution of the Hence the solution of the equation is equation is

( 3, - 2 ) ( 3, - 2 )

ELIMINATION METHODELIMINATION METHOD

In this method, we eliminate one of In this method, we eliminate one of the two variables to obtain an the two variables to obtain an equation in one variable which can equation in one variable which can easily be solved. Putting the value of easily be solved. Putting the value of this variable in any of the given this variable in any of the given equations, The value of the other equations, The value of the other variable can be obtained.variable can be obtained.

For example: we want to solve, For example: we want to solve,

3x + 2y = 113x + 2y = 11

2x + 3y = 42x + 3y = 4

Let 3x + 2y = 11 --------- (i)Let 3x + 2y = 11 --------- (i)

2x + 3y = 4 ---------(ii)2x + 3y = 4 ---------(ii)

Multiply 3 in equation (i) and 2 in equation Multiply 3 in equation (i) and 2 in equation (ii) and subtracting eq. iv from iii, we get(ii) and subtracting eq. iv from iii, we get

9x + 6y = 33 ------ (iii)9x + 6y = 33 ------ (iii)

4x + 6y = 8 ------- (iv)4x + 6y = 8 ------- (iv)

5x = 25 5x = 25

=> x = 5=> x = 5

putting the value of y in equation (ii) we putting the value of y in equation (ii) we get,get,

2x + 3y = 42x + 3y = 4

2 x 5 + 3y = 42 x 5 + 3y = 4

10 + 3y = 410 + 3y = 4

3y = 4 – 103y = 4 – 10

3y = - 63y = - 6

y = - 2y = - 2

Hence, x = 5 and y = -2Hence, x = 5 and y = -2