Question

Show that the system of equations has no solution.

Answer 1

The given system of equations can be written as
$$\begin{gathered} 6x+5y-11=.....\left(i\right) \\ 9x+\frac{15}{2}y-21=0.....\left(ii\right) \end{gathered}$$
This system is of the form
$$\begin{gathered} a_{1}x+b_{1}y+c_1=0 \\ {a}_{2}x+b_{2}y+c_2=0 \end{gathered}$$
where $a_1=6,b_1=5,c_1=-11\mathrm{and}{a}_2=9,b_2=\frac{15}{2},c_2=-21$
Now,
$$\begin{gathered} \frac{a_1}{a_2}=\frac{6}{9}=\frac{2}{3} \\ \frac{b_1}{b_2}=\frac{5}{{\displaystyle \frac{15}{2}}}=\frac{2}{3} \\ \frac{c_1}{c_2}=\frac{-11}{-21}=\frac{11}{21} \end{gathered}$$
Since, $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$, therefore the given system has no solution.