This document discusses methods for solving systems of linear equations in two variables, including graphical and algebraic methods. Graphical methods involve plotting the lines defined by each equation on a graph and finding their point of intersection. Algebraic methods covered are elimination by substitution, elimination by equating coefficients, and cross-multiplication. An example using substitution to solve the system x + 2y = -1 and 2x - 3y = 12 is shown, finding the solution (3, -2).

1. LINEARLINEAR
EQUATION INEQUATION IN
TWO VARIABLESTWO VARIABLES

2. System of equations orSystem of equations or
simultaneous equations –simultaneous equations –
A pair of linear equations in twoA pair of linear equations in two
variables is said to form a system ofvariables 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 equationsForm a system of two linear equations
in variables x and y.in variables x and y.

3. The general form of a linear equationThe general form of a linear equation
inin two variables x and y istwo variables x and y is
ax + by + c = 0 , a and b is not equalax + by + c = 0 , a and b is not equal
to zero, whereto zero, where
a, b and c being real numbers.a, b and c being real numbers.
A solution of such an equation is a pairA solution of such an equation is a pair
of values, One for x and the other for y,of values, One for x and the other for y,
Which makes two sides of the equationWhich makes two sides of the equation
equal.equal.
Every linear equation in two variables hasEvery linear equation in two variables has
infinitely many solutions which can beinfinitely many solutions which can be
represented on a certain line.

4. GRAPHICAL SOLUTIONS OF AGRAPHICAL SOLUTIONS OF A
LINEAR EQUATIONLINEAR EQUATION
Let us consider the following system of twoLet us consider the following system of two
simultaneous linear equations in twosimultaneous linear equations in two
variable.variable.
2x – y = -12x – y = -1
3x + 2y = 93x + 2y = 9
Here we assign any value to one of the twoHere we assign any value to one of the two
variables and then determine the value ofvariables and then determine the value of
the other variable from the giventhe other variable from the given
equation.equation.

5. 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
Y = 9-3xY = 9-3x
X 0 2
Y 1 5
X 3 -1
Y 0 6

6. XX
Y
Y
(2,5)
(-1,6)
(0,3)(0,1)
X=
1
Y=3

7. ALGEBRAIC METHODS OF SOLVINGALGEBRAIC METHODS OF SOLVING
SIMULTANEOUS LINEAR EQUATIONSSIMULTANEOUS LINEAR EQUATIONS
The most commonly used algebraicThe most commonly used algebraic
methods of solving simultaneous linearmethods of solving simultaneous linear
equations in two variables areequations in two variables are
• Method of elimination by substitution.Method of elimination by substitution.
• Method of elimination by equating theMethod of elimination by equating the
coefficient.coefficient.
• Method of Cross - multiplication.Method of Cross - multiplication.

8. ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION
STEPSSTEPS
Obtain the two equations. Let the equations beObtain 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) andChoose either of the two equations, say (i) and
find the value of one variable , say ‘y’ in termsfind the value of one variable , say ‘y’ in terms
of x.of x.
Substitute the value of y, obtained in theSubstitute the value of y, obtained in the
previous step in equation (ii) to get an equationprevious step in equation (ii) to get an equation
in x.in x.

9. ELIMINATION BY SUBSTITUTIONELIMINATION BY SUBSTITUTION
Solve the equation obtained in theSolve 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 theSubstitute 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)

10. SUBSTITUTION METHODSUBSTITUTION METHOD
x + 2y = -1x + 2y = -1
x = -2y -1 ------- (iii)x = -2y -1 ------- (iii)
Substituting the value of x inSubstituting 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 ,

11. SUBSTITUTIONSUBSTITUTION
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) – 1x = - 2 x (-2) – 1
= 4 – 1= 4 – 1
= 3= 3
Hence the solution of the equation isHence the solution of the equation is
( 3, - 2 )( 3, - 2 )

12. ELIMINATION METHODELIMINATION METHOD
In this method, we eliminate one of theIn this method, we eliminate one of the
two variables to obtain an equation in onetwo 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 ofPutting the value of this variable in any of
the given equations, The value of the otherthe 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

13. 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)Multiply 3 in equation (i) and 2 in equation (ii)
and subtracting eq. iv from iii, we getand subtracting eq. iv from iii, we get
9x + 6y = 33 ------ (iii)9x + 6y = 33 ------ (iii)
4x + 6y = 8 ------- (iv)4x + 6y = 8 ------- (iv)
5x = 255x = 25
=> x = 5=> x = 5

14. 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