Question

Find the value of ‘k’ which each of the following system of equations has infinite number of solutions

$$\begin{gathered} 2x-y+8=0 \\ 4x-ky+16=0 \end{gathered}$$

Answer 1

Step1: Applying and Analysing

The given system of equation is

$$\begin{gathered} 2x-y+8=0 \\ 4x-ky+16=0 \end{gathered}$$

The given system of equation is of the form

$$\begin{gathered} a_{1}x+b_{1}y+c_1=0 \\ {a}_{2}x+b_{2}y+c_2=0 \end{gathered}$$

where,

$$\begin{gathered} a_1=2,b_1=-1,c_1=8 \\ {a}_2=4,b_2=-k,c_2=16 \end{gathered}$$

Step 2: Equating the coefficients

For infinitely many solutions, we have

$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ i.e,\frac{2}{4}=\frac{-1}{-k}=\frac{8}{16} \\ \frac{1}{2}=\frac{1}{k}=\frac{1}{2} \\ \frac{1}{k}=\frac{1}{2} \\ k=2 \end{gathered}$$

Hence the given system of equation will have infinitely many solutions if k=2.