This document discusses methods for solving systems of two linear equations with two unknown variables. It explains that a system of two linear equations can be represented graphically by finding the point of intersection between the lines defined by each equation. Algebraically, there are three main methods covered: elimination by substitution, elimination by equating coefficients, and cross-multiplication. An example using the substitution method is worked through step-by-step to find the solution (3, -2).

1. LINEARLINEAR
EQUATION INEQUATION IN
TWOTWO
VARIABLESVARIABLESBY CHRISTOPHER
SELVARAJ

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 equation inThe general form of a linear equation in
two variables x and y istwo variables x and 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 ofA solution of such an equation is a pair of
values, one for x and the other for y, whichvalues, 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 hasEvery linear equation in two variables has
infinitely many solutions which can beinfinitely many solutions which can be
represented on a certain line.represented on a certain line.

4. GRAPHICAL SOLUTIONS OF AGRAPHICAL SOLUTIONS OF A
LINEAR EQUATIONLINEAR EQUATION
Let us consider the following system ofLet us consider the following system of
two simultaneous linear equations in twotwo simultaneous 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 given equation.the other variable from the given 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
9- 3x9- 3x
Y = -------Y = -------
22
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

8. ALGEBRAIC METHODS OFALGEBRAIC METHODS OF
SOLVING SIMULTANEOUSSOLVING SIMULTANEOUS
LINEAR EQUATIONSLINEAR 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 substitutionMethod of elimination by substitution
Method of elimination by equating theMethod of elimination by equating the
coefficientcoefficient
Method of Cross- multiplicationMethod of Cross- multiplication

9. 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 xof 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 xin x

10. 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)

11. 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 ,

12. 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 )

14. 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

15. 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) andMultiply 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 = 255x = 25
=> x = 5=> x = 5

16. 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