Question

Verify if the following simultaneous equations are consistent or inconsistent.
I. $2x+5y=17$, $5x+3y=14$
II. $2x+6y=7$, $6x+18y=10$

Answer 1

(i) Comparing $2x+5y=17$ with $a_{1}x+b_{1}y+c_1=0$, we get, $a_1=2,b_1=5,c_1=-17$

and $5x+3y=14$ with $a_{2}x+b_{2}y+c_2=0$, we get, $a_2=5,b_2=3,c_2=-14$

Here,

$\frac{a_1}{a_2}=\frac{2}{5}$, $\frac{b_1}{b_2}=\frac{5}{3}$

$\because \frac{2}{5}\ne \frac{5}{3}$

The equations are consistent.

(ii) Comparing $2x+6y=7$ with $a_{1}x+b_{1}y+c_1=0$,we get,

$a_1=2,b_1=6,c_1=-7$`

and$6x+18y=10$ with $a_{2}x+b_{2}y+c_2=0$,we get,

$a_2=6,b_2=18,c_2=-10$

Here,

$\frac{a_1}{a_2}=\frac{2}{6}$,$\frac{b_1}{b_2}=\frac{6}{18}$

$\frac{c_1}{c_2}=\frac{-7}{-10}$

$\because \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

$\therefore$ the equations are inconsistent.