Question

Match the following:
$\begin{array}{cc}A.y=-\frac{3}{2}-\frac{x}{2};x=-3-2y& \left(i\right)Inconsistentpair\\ B.5x=6-3y;9y=12-15x& \left(ii\right)Dependentpair\\ C.y=\frac{2}{7}-\frac{x}{7};y=\frac{18}{5}-\frac{x}{5}& \left(iii\right)Uniquesolution\end{array}$

• A. A-i, B-iii, C-ii
• B. A-ii, B-i, C-iii
• C. A-ii, B-iii, C-i
• D. A-iii, B-i, C-ii

Answer 1

The correct option is B

A-ii, B-i, C-iii

Write each of the pairs of equations in the standard form where
$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$
Inconsistent pair: $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
Dependent pair: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
Unique solution:$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

$A.y=-\frac{3}{2}-\frac{x}{2}\Rightarrow \frac{x}{2}+y+\frac{3}{2}=0x=-3-2y\Rightarrow x+2y+3=0Here\frac{a_1}{a_2}=\frac{1}{2},\frac{b_1}{b_2}=\frac{1}{2},\frac{c_1}{c_2}=\frac{1}{2}$
Hence,$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$. Thus, it is a dependent pair.

$B.5x=6-3y\Rightarrow 5x+3y-6=0$
$9y=12-15x\Rightarrow 15x+9y-12=0$
Here $\frac{a_1}{a_2}=\frac{1}{3},\frac{b_1}{b_2}=\frac{1}{3},\frac{c_1}{c_2}=\frac{1}{2}$
Hence,$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$. Thus, it is an inconsistent pair.

$C.y=\frac{2}{7}-\frac{x}{7}\Rightarrow \frac{x}{7}+y-\frac{2}{7}=0$
$y=\frac{18}{5}-\frac{x}{5}\Rightarrow \frac{x}{5}+y-\frac{18}{5}=0$
Here $\frac{a_1}{a_2}=\frac{1}{6},\frac{b_1}{b_2}=\frac{5}{7}$
Hence, $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$. Thus, it is a consistent pair.