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 of variables is said to form a system of simultaneous linear equations.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 equations Form a system of two linear equations in variables x and y.in variables x and y.

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

ax + by + c = 0 , a =/= 0, b=/=0, ax + by + c = 0 , a =/= 0, b=/=0, wherewhere

a, b and c being real numbers.a, b and c being real numbers.A solution of such an equation is a pair of A solution of such an equation is a pair of values, one for x and the other for y, which values, one for x and the other for y, which makes two sides of the equation equal. makes two sides of the equation equal.

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

GRAPHICAL SOLUTIONS OF A GRAPHICAL SOLUTIONS OF A LINEAR EQUATIONLINEAR EQUATION

Let us consider the following system of Let us consider the following system of two simultaneous linear equations in two two simultaneous linear equations in two variable.variable. 2x – y = -12x – y = -13x + 2y = 93x + 2y = 9

Here we assign any value to one of the two Here we assign any value to one of the two variables and then determine the value of variables and then determine the value of the other variable from the given equation.the other 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 – 3x2y = 9 – 3x

9- 3x9- 3xY = -------Y = ------- 22

X 0 2

Y 1 5

X 3 -1

Y 0 6

XX’

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 linear equations in two variables areequations in two variables areMethod of elimination by substitutionMethod of elimination by substitutionMethod of elimination by equating the Method of elimination by equating the coefficientcoefficientMethod of Cross- multiplicationMethod of Cross- multiplication

ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTIONSTEPSSTEPS

Obtain the two equations. Let the equations be Obtain the two equations. Let the 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) and Choose either of the two equations, say (i) and find the value of one variable , say ‘y’ in terms find the value of one variable , say ‘y’ in terms of xof xSubstitute the value of y, obtained in the Substitute the value of y, obtained in the previous step in equation (ii) to get an equation previous step in equation (ii) to get an equation in xin 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 the Substitute the value of x and get the value of y.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 = -1x = -2y -1 ------- (iii)x = -2y -1 ------- (iii)Substituting the value of x in equation Substituting the value of x in equation (ii), we get(ii), we get2x – 3y = 122x – 3y = 122 ( -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 getPutting the value of y in eq (iii), we getx = - 2y -1x = - 2y -1x = - 2 x (-2) – 1 x = - 2 x (-2) – 1

= 4 – 1 = 4 – 1 = 3= 3

Hence the solution of the equation is Hence the solution of the equation is ( 3, - 2 ) ( 3, - 2 )

ELIMINATION METHODELIMINATION METHOD

In this method, we eliminate one of the In this method, we eliminate one of the two variables to obtain an equation in one two variables to obtain an equation in one variable which can easily be solved. variable which can easily be solved. Putting the value of this variable in any of Putting the value of this variable in any of the given equations, the value of the other the given 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 (ii) and Multiply 3 in equation (i) and 2 in equation (ii) and

subtracting eq iv from iii, we getsubtracting 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 get,putting the value of y in equation (ii) we get,

2x + 3y = 42x + 3y = 42 x 5 + 3y = 42 x 5 + 3y = 410 + 3y = 410 + 3y = 43y = 4 – 103y = 4 – 103y = - 63y = - 6y = - 2y = - 2

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

Prepared byPrashant.

Tiwari.X-A