Question

Without actually solving the simultaneous equations given below,
decide which simultaneous equations have unique solutions ,
no solution or infinitely many solutions.

$\left(\mathrm{i}\right)3x+5y=16;4x-y=6$

Answer 1

$$\begin{gathered} 3x+5y=16 \\ 4x-y=6 \end{gathered}$$

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

$$\begin{gathered} a_1=3b_1=5c_1=16 \\ {a}_2=4b_2=-1c_2=6 \\ \therefore \frac{a_1}{a_2}=\frac{3}{4};\frac{b_1}{b_2}=\frac{5}{-1}=-5\mathrm{and}\frac{c_1}{c_2}=\frac{16}{6}=\frac{8}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{i}.\mathrm{e}.,\frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \therefore \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{have}\mathrm{unique}\mathrm{solutions}. \end{gathered}$$