OBJECTIVE
To verify the conditions of consistency/
inconsistency for a pair of linear
equations in two variables by graphical
method.
MATERIAL REQUIRED
Graph papers, pencil, eraser,
cardboard, glue.
METHOD OF CONSTRUCTION
1. Take a pair of linear equations in two variables of the form
a1
x + b1
y + c1
= 0 (1)
a2
x + b2
y + c2
= 0, (2)
where a1
, b1
, a2
, b2
, c1
and c2
are all real numbers; a1
, b1
, a2
and b2
are not
simultaneously zero.
There may be three cases :
Case I : 1 1
2 2
a b
a b
≠
Case II: 111
222
abc
abc = =
Case III: 111
222
abc
abc
= ≠
2. Obtain the ordered pairs satisfying the pair of linear equations (1) and (2)
for each of the above cases.
3. Take a cardboard of a convenient size and paste a graph paper on it. Draw
two perpendicular lines X′OX and YOY′ on the graph paper (see Fig. 1).
Plot the points obtained in Step 2 on different cartesian planes to obtain
different graphs [see Fig. 1, Fig. 2 and Fig.3].
DEMONSTRATION
Case I: We obtain the graph as shown in Fig. 1. The two lines are intersecting
at one point P. Co-ordinates of the point P (x,y) give the unique solution for the
pair of linear equations (1) and (2).
Therefore, the pair of linear equations with 1 1
2 2
a b
a b
≠ is consistent and has the
unique solution.
Case II: We obtain the graph as shown in Fig. 2. The two lines are coincident.
Thus, the pair of linear equations has infinitely many solutions.
Therefore, the pair of linear equations with 111
222
abc
abc = = is also consistent as
well as dependent.
Case III: We obtain the graph as shown in Fig. 3. The two lines are parallel to
each other.
This pair of equations has no solution, i.e., the pair of equations with
111
222
abc
abc
= ≠ is inconsistent.
OBSERVATION
1. a1
= __________, a2
= __________,
b1
= __________, b2
= __________,
c1
= __________, c2
= __________,
SO, 1
2
a
a = ____________, 1
2
b
b = ____________,
1
2
c
c
= _______________
.
1
2
a
a
1
2
b
b
1
2
c
c
Case I, II or III Type of lines Number of
solution
Conclusion
Consistent/
inconsistent/
dependent
APPLICATION
Conditions of consistency help to check whether a pair of linear equations have
solution (s) or not.
In case, solutions/solution exist/exists, to find whether the solution is unique
or the solutions are infinitely many.
Answers
Answer:
Step-by-step explanation:
OBJECTIVE: To verify the conditions of consistency/ inconsistency for a pair of linear equations in two variables by graphical method.
MATERIAL REQUIRED: Graph papers, pencil, eraser, cardboard and glue.
METHOD OF CONSTRUCTION: 1. Take a pair of linear equations in two variables of the form a1 x + b1 y + c1 = 0 (1) a2 x + b2 y + c2 = 0 (2) where a1 , b1 , a2 , b2 , c1 and c2 are all real numbers; a1 , b1 , a2 and b2 are not simultaneously zero.
There may be three cases:
Case I : a 1 /a2 = b1 / b2
Case II: a 1 /a2 = b1 / b2 = c1/c2
Case III: a 1 /a2 = b1 / b2 = c1/c2
2. Obtain the ordered pairs satisfying the pair of linear equations (1) and (2) for each of the above cases. 3. Take a cardboard of a convenient size and paste a graph paper on it. Draw two perpendicular lines X′OX and YOY′ on the graph paper. Plot the points obtained in Step 2 on different Cartesian planes to obtain different graphs.
Topic
Linear equations in two variables
Enrichment Activity No.1
Month: April
Class
X
Subject
Mathematics
Name
Date : 30.04.21