Question

In the following system of equation determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it.
$6x+5y=11$

$9x+\frac{15}{2}y=21$

Answer 1

The given system of equations may be written as
$6x+5y-11=0$

$9x+\frac{15}{2}y-21=0$

The given system of equations is of the form $a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$

where, $a_1=6,b_1=5,c_1=-11,a_2=9,b_2=\frac{15}{2},c_2=-21$

We have, $\frac{a_1}{a_2}=\frac{6}{9}=\frac{2}{3},\frac{b_1}{b_2}=\frac{5}{15/2}=\frac{2}{3}$ and $\frac{c_1}{c_2}=\frac{-11}{-21}=\frac{11}{21}$

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

So, the given system of equations has no solution i.e., it is in-consistent.