Question

Which of the following pairs of equations is inconsistent?

• A. $2x+3y=15;4x+7y=11$
• B. $4x+y=12;8x+2y=24$
• C. $x+2y=5;2x+4y=15$
• D. $11y+3x=1;5y+2x=7$

Answer 1

The correct option is C $x+2y=5;2x+4y=15$

A pair of linear equations in two variables is said to be inconsistent when:
$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Consider the equations given in option A.
$2x+3y=15$
$4x+7y=11$
$Here,a_1=2,b_1=3and{c}_1=-15$
$Also,a_2=4,b_2=7and{c}_2=-11$
Here, $\frac{2}{4}\ne \frac{3}{7}$
$\Rightarrow \frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
$\therefore$ Equations in option A have unique solution and are consistent.

Consider equations given in option B.
$\frac{4}{8}=\frac{1}{2}=\frac{12}{24}\Rightarrow \frac{1}{2}=\frac{1}{2}=\frac{1}{2}$
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
$\therefore$ Equations in option B have infinite solutions and are consistent.

Consider equations given in option C.
$\frac{1}{2}=\frac{2}{4}\ne \frac{5}{15}$
$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
$\therefore$ Equations in option C have no solution and are inconsistent.


Consider equations given in option D.
$\frac{3}{2}\ne \frac{11}{5}$
$\Rightarrow \frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
$\therefore$ Equations in option D have a unique solution and are consistent.