Question

two rails are represented by the equations x+2y-4=0 and 2x+4y-12=0. solve it algebrically and will the rails cross each other?

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{are}:- \\ x+2y=4........\left(1\right) \\ \mathrm{and}2x+4y=12 \\ \mathrm{or}2\left(x+2y\right)=12 \\ \mathrm{or}x+2y=6........\left(2\right) \\ \mathrm{Comparing}\mathrm{the}\mathrm{above}2\mathrm{equations}\mathrm{with} \\ {a}_{1}x+b_{1}y=c_1\mathrm{and}{a}_{2}x+b_{2}x=c_2,\mathrm{we}\mathrm{get}, \\ {a}_1=1;b_1=2;c_1=4 \\ {a}_2=1;b_2=2;c_2=6 \\ \mathrm{Now},\frac{a_1}{a_2}=1 \\ \frac{b_1}{b_2}=\frac{2}{2}=1 \\ \frac{c_1}{c_2}=\frac{4}{6}=\frac{2}{3} \\ \mathrm{Since},\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}, \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{system}\mathrm{of}\mathrm{equations}\mathrm{has}\mathrm{no}\mathrm{solution}. \\ \mathrm{Also},\mathrm{the}2\mathrm{straight}\mathrm{lines}\mathrm{represented}\mathrm{by}\mathrm{the}\mathrm{above}2\mathrm{equations}\mathrm{will}\mathrm{be}\mathrm{parallel}\mathrm{to}\mathrm{each}\mathrm{other}. \\ \mathrm{Hence}\mathrm{the}\mathrm{two}\mathrm{rails}\mathrm{will}\mathrm{not}\mathrm{cross}\mathrm{each}\mathrm{other}. \end{gathered}$$