Question

Q1. State the condition of solvability for :
(a) a consistent system of equations.
(b) an inconsistent system of equations.
(c) a dependent system of equations.

Answer 1

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{following}\mathrm{system}\mathrm{of}\mathrm{linear}\mathrm{equations}: \\ {\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0.....\left(1\right) \\ {\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0.......\left(2\right) \\ \left(\mathrm{a}\right).\mathrm{For}\mathrm{consitent}\mathrm{system}\mathrm{of}\mathrm{equations}: \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}\ne \frac{{\mathrm{b}}_1}{{\mathrm{b}}_2} \\ \left(\mathrm{b}\right).\mathrm{For}\mathrm{inconsistent}\mathrm{system}\mathrm{of}\mathrm{equations} \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\ne \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \\ \left(\mathrm{c}\right).\mathrm{For}\mathrm{dependent}\mathrm{system}\mathrm{of}\mathrm{equations} \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \end{gathered}$$