Question

The following pair of linear equations is consistent or inconsistent?

$$\begin{gathered} 2\mathrm{x}–3\mathrm{y}=8 \\ 4\mathrm{x}–6\mathrm{y}=9 \end{gathered}$$

Answer 1

Given:

$$\begin{gathered} 2\mathrm{x}–3\mathrm{y}=8 \\ 4\mathrm{x}–6\mathrm{y}=9 \end{gathered}$$

For any pair of linear equations,

$$\begin{gathered} {\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0 \\ {\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0 \end{gathered}$$

• $\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}\ne \frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}$ (Intersecting Lines / unique solution / consistent)
• $\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}$ (Coincident Lines / Infinitely many Solutions / consistent)
• $\frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\ne \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}$ (Parallel Lines / No solution / inconsistent)

Write the given equation in standard form,

$$\begin{gathered} \Rightarrow 2\mathrm{x}–3\mathrm{y}-8=0 \\ \Rightarrow 4\mathrm{x}–6\mathrm{y}-9=0 \end{gathered}$$

Where,

$$\begin{gathered} {\mathrm{a}}_1=2,{\mathrm{b}}_1=-3,{\mathrm{c}}_1=-8 \\ {\mathrm{a}}_2=4,{\mathrm{b}}_2=-6,{\mathrm{c}}_2=-9 \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{2}{4}=\frac{1}{2},\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{-3}{-6}=\frac{1}{2},\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2}=\frac{-8}{-9}=\frac{8}{9} \\ \because \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\ne \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \end{gathered}$$

Hence, the given pair of linear equations is inconsistent.