Page 18 - Mathematics Class - X

ACTIVITY 2.2

OBJECTIVE
To verify the conditions of consistency/inconsistency for a pair of linear equations in two variables by graphical
method.

MATERIALS REQUIRED

Graph papers Eraser Glue
Pencil Cardboard

PRE-REQUISITE KNOWLEDGE
1. Concept of linear equation 2. Pair of linear equations

3. Plotting a graph of linear equation

THEORY
1. An equation of the form ax + by + c = 0, is called linear equation in two variables.
2. The graph of a linear equation say ax + by + c = 0 in two variables is always a straight line.
3. Two linear equations in the same two variables are said to form a pair of linear equations in two variables.
The most general form of a pair of linear equations is
a x + b y + c = 0 and a x + b y + c = 0
1 1 1 2 2 2
4. Different cases that can happen with the graph of a pair of linear equations in two variables.
Case 1 If the lines intersect at a point, then that point gives the unique solution of the two equations.
In this case, the pair of equations is said to be consistent.

a b
where 1 ≠ 1 .
a 2 b 2
Case 2 If the lines coincide, then there are infinitely many solutions where each point on the line is a
solution. In this case, the pair of equations is said to be consistent (dependent).
a b c
where, 1 = 1 = 1 .
a 2 b 2 c 2
Case 3 If the lines are parallel, i.e. they do not intersect each other at any point, then the pair of equations
has no solution. In this case, the pair of equations is said to be inconsistent.
a b c
where, 1 = 1 ≠ 1 .
a 2 b 2 c 2

PROCEDURE
1. Take a pair of linear equations in two variables, say
a x + b y + c = 0 ...(i)
1 1 1
and a x + b y + c = 0 ...(ii)
2 2 2
where, a , b , a , b , c and c are all real numbers; a , b , a and b are not simultaneously zero.
1 1 2 2 1 2 1 1 2 2

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