Answer the following questions related to a pair of linear equations in two variables…

MATHEMATICS · Multi-part · CBSE · Mathematics · Pair of Linear Equations in Two Variables · Pair of Linear Equations in Two Variables

Problem

Answer the following questions related to a pair of linear equations in two variables

a_1x + b_1y + c_1 = 0

and

a_2x + b_2y + c_2 = 0

. (i) Explain the meaning of the following terms with respect to the above pair of equations: (A) consistent and independent, (B) inconsistent, (C) dependent. (ii) Show that (A) the pair is consistent and independent if

\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}

, (B) the pair is inconsistent if

\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}

, (C) the pair is dependent if

\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}

. Give a brief graphical reason for each case. (iii) Construct any two distinct linear equations that represent coincident lines. Reduce both equations to the simplest form, verify the equality of the three ratios obtained in part (ii)(C), and draw a neat, labelled rough sketch of the common line on a Cartesian plane.

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