Question

Find the value of p for which the given simultaneous equations have unique solution :
$\left(\mathrm{ii}\right)8x-py+7=0;4x-2y+3=0$

Answer 1

$$\begin{gathered} 8x-py+7=0 \\ 4x-2y+3=0 \end{gathered}$$

These equations can be written as follows:

$$\begin{gathered} 8x-py=-7 \\ 4x-2y=-3 \end{gathered}$$

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

$$\begin{gathered} a_1=8b_1=-p{c}_1=-7 \\ {a}_2=4b_2=-2c_2=-3 \\ \therefore \frac{a_1}{a_2}=\frac{8}{4}=2;\frac{b_1}{b_2}=\frac{-p}{-2}=\frac{p}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{-7}{-3}=\frac{7}{3} \end{gathered}$$

The condition for simultaneous equations to have a unique solution is
$$\begin{gathered} \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \therefore 2\ne \frac{p}{2} \\ \mathrm{i}.\mathrm{e}.,p\ne 4 \\ \therefore \mathrm{The}\mathrm{given}\mathrm{simultaneous}\mathrm{equations}\mathrm{will}\mathrm{have}\mathrm{a}\mathrm{unique}\mathrm{solution}\mathrm{for}\mathrm{all}\mathrm{values}\mathrm{of}p,\mathrm{except}p=4. \end{gathered}$$