Question

Are the following pair of linear equations consistent?

$\frac{3}{5}x-y=\frac{1}{2};\frac{1}{5}x-3y=\frac{1}{6}$

Answer 1

The given pair if linear equation can be written as

$\frac{3}{5}x-y-\frac{1}{2}=0$ and $\frac{1}{5}x-3y-\frac{1}{6}$

Here, $a_1=\frac{3}{5},b_1=-1,c_1=-\frac{1}{2}$

And, $a_2=\frac{1}{5},b_2=-3,c_2=-\frac{1}{6}$

$\therefore \frac{a_1}{a_2}=\frac{\frac{3}{5}}{\frac{1}{5}}=3$ and $\frac{b_1}{b_2}=\frac{-1}{-3}=\frac{1}{3}$ and $\frac{c_1}{c_2}=\frac{\frac{1}{2}}{\frac{1}{6}}=3$

Hence, $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

Thus, the given pair of linear equations consistent.