Question

Find the value of p for which the given simultaneous equations have unique solution :
$\left(\mathrm{i}\right)3x+y=10;9x+py=23$

Answer 1

$$\begin{gathered} 3x+y=10 \\ 9x+py=23 \end{gathered}$$

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

$$\begin{gathered} a_1=3b_1=1c_1=10 \\ {a}_2=9b_2=p{c}_2=23 \\ \therefore \frac{a_1}{a_2}=\frac{3}{9}=\frac{1}{3};\frac{b_1}{b_2}=\frac{1}{p}\mathrm{and}\frac{c_1}{c_2}=\frac{10}{23} \end{gathered}$$

The condition for simultaneous equations to have a unique solution is
$$\begin{gathered} \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \therefore \frac{1}{3}\ne \frac{1}{p} \\ \mathrm{i}.\mathrm{e}.,p\ne 3 \\ \therefore \mathrm{The}\mathrm{given}\mathrm{simultaneous}\mathrm{equations}\mathrm{will}\mathrm{have}\mathrm{a}\mathrm{unique}\mathrm{solution}\mathrm{for}\mathrm{all}\mathrm{values}\mathrm{of}p,\mathrm{except}\mathrm{when}p=3. \end{gathered}$$