LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN LINEAR EQUATIONS IN TWO VARIABLESTWO VARIABLES

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.

GENERAL FORMGENERAL FORM The general form of a linear equation in two The general form of a linear equation in two

variables xand y is variables xand y is

ax + by + c = 0 , a =/= 0, b=/=0, whereax + by + c = 0 , a =/= 0, b=/=0, where

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.

TO CHECK IF THE PAIR OF TO CHECK IF THE PAIR OF EQUATIONS HAVE SOLUTIONEQUATIONS HAVE SOLUTION

Firstly we have to know that whether the equations Firstly we have to know that whether the equations can be solved or not. For this we have the rules can be solved or not. For this we have the rules shown below :-shown below :-

1. a1. a11 =/= b =/= b11 (UNIQUE SOLUTION) (UNIQUE SOLUTION)

aa22 b b22

2. a2. a11 = b = b11 = c = c11 (INFINITE SOLUTIONS) (INFINITE SOLUTIONS)

aa22 b b22 c c22

3. a3. a11 = b = b11 =/= c =/= c11 (NO SOLUTION) (NO SOLUTION)

aa22 b b22 c c22

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 are

*Method of substitution*Method of substitution

*Method of elimination *Method of elimination

*Method of Cross- multiplication*Method of Cross- multiplication

SUBSTITUTION METHODSUBSTITUTION METHOD

STEPSSTEPS

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 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 equation previous step in equation (ii) to get an equation in xin x


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)


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 ,


Putting the value of y in eq (iii), we getPutting the value of y in eq (iii), we get

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 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.

We eliminate one variable first , to get a We eliminate one variable first , to get a linear equation in one variable. linear equation in one variable.

Step 1. first multiply both the equation by some suitable non-zero constants to make the coefficients of one variable numerically equal.Step 2. then add or subtract one equation from the other so that one variable gets eliminated. If you get an equation in one variable, go to step Step 3. solve equation in one variable so obtained to get its value.For example: we want to solve, 3x + 2y = 11 2x + 3y = 4

ELIMINATION METHOD

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 eqn. iv from iii, we getsubtracting eqn. 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 = 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

CROSS MULTIPLICATION CROSS MULTIPLICATION METHODMETHOD

This is a method very useful for solving the This is a method very useful for solving the linear equation in two variableslinear equation in two variables

Let us consider two equations:-Let us consider two equations:-aa11x + bx + b11y + cy + c11 = 0 = 0

aa22x + bx + b22y + cy + c22 = 0 = 0

x y 1x y 1

bb11 c c11 a a11 bb11

bb22 c c22 a a22 bb22


x = y = 1x = y = 1

bb11cc22 – b – b22cc1 1 c c11aa22 – c – c22aa11 a a11bb22 – a – a22bb11

By this way the equations are solved and By this way the equations are solved and the values are obtained.the values are obtained.

We have to put the values of the known We have to put the values of the known and get the values of the unknown.and get the values of the unknown.

We can write this as given below alsoWe can write this as given below also


x = bx = b11cc22 - b - b22cc11

aa11bb22 - a - a22bb11

y = cy = c11aa22 - c - c22aa11

aa11bb22 - a - a22bb11

REDUCIBLE EQUATIONSREDUCIBLE EQUATIONS

The equations which cannot be solved The equations which cannot be solved simply are converted to the reduced forms simply are converted to the reduced forms and then solved such as :-and then solved such as :-

• 2/x + 3/y = 13 2/x + 3/y = 13 2(1/x) + 3(1/y) = 13 2(1/x) + 3(1/y) = 13• 5/x – 4/y = -2 5/x – 4/y = -2 5(1/x) – 4(1/y) = -2 5(1/x) – 4(1/y) = -2

Let (1/x) = p & (1/y) = q, then the Let (1/x) = p & (1/y) = q, then the equations :- 2p + 3q = 13 & 5p – 4q = -2 equations :- 2p + 3q = 13 & 5p – 4q = -2 can be solved by any of the three methods can be solved by any of the three methods mentioned above.mentioned above.

FOMATIVE ASSESSMENT(MCQ)1. Which of the following is the solution of the pair of linear equations 3x – 2y = 0, 5y – x = 0

(a) (5, 1) (b) (2, 3) (c) (1, 5) (d) (0, 0)

2. One of the common solution of ax + by = c and y-axis is _____

(a) (0, c/b) (b) (0,b/c ) (c) , 0 , (c/ b ) (d) (0, c/ b)

3. If the value of x in the equation 2x – 8y = 12 is 2 then the corresponding value of y will be

(a) –1 (b) +1 (c) 0 (d) 2

ACTIVITYACTIVITYTo verify graphicallyTo verify graphicallyThat the pair of linear equations That the pair of linear equations x+y-5=0 ,2x+2y-6=0 x+y-5=0 ,2x+2y-6=0 in which in which a1/a2=b1/b2 =/= c1/c2 gives a pair of parallel straight lines.a1/a2=b1/b2 =/= c1/c2 gives a pair of parallel straight lines.

x + y = 5 …..(1)x + y = 5 …..(1)

2x +2y = 6…(2)2x +2y = 6…(2)

SOLUTION FOR X+Y=5SOLUTION FOR X+Y=5

XX 00 55

YY 55 00

SOLUTION FOR 2X+2Y=6SOLUTION FOR 2X+2Y=6

XX 00 33

YY 33 00

After plotting the points on graph we getAfter plotting the points on graph we get

Which represent the pair of parallel lines

LINEAR EQUATIONS IN TWO VARIABLES

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