ACTIVITY 03SOLVING

EQUATIONS USING GRAPHICAL METHOD

CLASS X

OBJECTIVE

TO VERIFY THE CONDITIONS OF CONSISTENCY/ INCONSISTENCY FOR A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES BY GRAPHICAL METHOD.

MATERIALS REQUIRED

GRAPH PAPERSPENCIL ERASERCARDBOARDGLUE

METHOD OF CONSTRUCTIONTAKE A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES OF THE FORM

a1 x + b1 y = c1; a2 x+ b2 y = c2

Where a1, b1, a2, b2, c1 and c2 are all real numbers and a1, b1, a2, b2 all ≠ 0.

METHOD OF CONSTRUCTION

There may be three cases:Case I: a1 ≠ b1 a2 b2Case II : a1 = b1 = c1 a2 b2 c2Case III : a1 = b1 ≠ c1 a2 b2 c2

METHOD OF CONSTRUCTION

Obtain the ordered pairssatisfying the pair of Linear equations (1) and(2) for each of the Abovecases.

METHOD OF CONSTRUCTIONTake 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 [ Fig.1]Plot the points obtained on different Cartesian planes to obtain different graphs. [Fig. 1, Fig. 2 and Fig. 3]

Y’ a2x+b2y=c2

X’ 0 X

a1x+ b1y = c1

FIG. 1 Y

FIG .2 Y a1x+ b1y = c1

X’ 0 X

a2x+ b2y = c2

Y’

Y

a2x+b2y=c2

X’ 0 X

a1x+b1y=c1

Y’

CASE 1: 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 a1 ≠ b1 IS CONSISTENT AND HAS a2 b2THE UNIQUE SOLUTION.

DEMONSTRATION

CASE II: WE OBTAIN THE GRAPH AS SHOWN IN FIG2. THE TWO LINES ARE COINCIDENT. THUS, THE PAIR OF LINEAR EQUATIONS HAS INFINITELY MANY SOLUTIONS. THEREFORE, THE PAIR OF LINEAR EQUATIONS WITH a1 = b1 = c1 IS CONSISTENT.

a2 b2 c2

DEMONSTRATION

OBSERVATION CASE III: WE OBTAIN THE GRAPH AS SHOWN IN FIG.3. THE TWO LINES ARE PARALLEL TO EACH OTHER.THE PAIR OF EQUATIONS HAS NO SOLUTION.i.e. THE PAIR OF EQUATIONS WITH a1 = b1 ≠ c1 a2 b2 c2IS INCONSISTENT.

OBSERVATION1. a1 = ___________ a2 = ________________

b1 = ___________ b2 = ________________

c1 = ___________ d1 = ________________

So,

a1 = _______ b1 = _______ c1 = _________ a2 b2 c2

a1

a2

b1

b2

c1 c2

Case I, II and III

Type of lines

Number of solution

Conclusion

CONCLUSION

Conditions of consistency help to check whether a pair of linear equations have solution[s] or not.In case, solutions exists, to find whether the solution is unique or the solutions are infinitely many.

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