Question

Find the value of k for which the given simultaneous equations have infinitely many solutions :
$\left(\mathrm{i}\right)4x+y=7;16x+ky=28$

Answer 1

$$\begin{gathered} 4x+y=7 \\ 16x+ky=28 \end{gathered}$$

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

$$\begin{gathered} a_1=4b_1=1c_1=7 \\ {a}_2=16b_2=k{c}_2=28 \\ \therefore \frac{a_1}{a_2}=\frac{4}{16}=\frac{1}{4};\frac{b_1}{b_2}=\frac{1}{k}\mathrm{and}\frac{c_1}{c_2}=\frac{7}{28}=\frac{1}{4} \end{gathered}$$

The condition for simultaneous equations to have infinitely many solutions is
$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \Rightarrow \frac{1}{4}=\frac{1}{k}=\frac{1}{4} \\ \Rightarrow \frac{1}{k}=\frac{1}{4} \\ \therefore k=4 \end{gathered}$$