Question

Without actually solving the simultaneous equations given below,
decide which simultaneous equations have unique solutions ,
no solution or infinitely many solutions.
$\left(\mathrm{iii}\right)3x-7y=15;6x=14y+10$

Answer 1

$$\begin{gathered} 3x-7y=15 \\ 6x=14y+10 \end{gathered}$$

These equations can be written as follows:

$$\begin{gathered} 3x-7y=15 \\ 6x-14y=10 \end{gathered}$$

Comparing the above equations with the equations $a_{1}x+b_{1}y=c_1\mathrm{and}{a}_{2}x+b_{2}y=c_2,\mathrm{we}\mathrm{get}$:

$$\begin{gathered} a_1=3b_1=-7c_1=15 \\ {a}_2=6b_2=-14c_2=10 \\ \therefore \frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2};\frac{b_1}{b_2}=\frac{-7}{-14}=\frac{1}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{15}{10}=\frac{3}{2} \end{gathered}$$

$$\begin{gathered} \mathrm{i}.\mathrm{e}.,\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2} \\ \therefore \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{have}\mathrm{no}\mathrm{solutions}. \end{gathered}$$