Question

Find the value of $k$ for which the given simultaneous equations have infinitely many solutions: $4x+y=7,16x+ky=28$.

Answer 1

$4x+y=7$ and $16x+ky=28$

Comparing the above equation with the equation $a_{1}x+b_{1}y=c_1$ and

$a_{2}x+b_{2}y=c_2$ respectively.

$a_1=4,b_1=1,c_1=7$,

$a_2=16,b_2=k,c_2=28$

Apply the condition for infinitely many solution.

$\frac{a_1}{a_2}=\frac{4}{16}=\frac{1}{4}$,

$\frac{b_1}{b_2}=\frac{1}{k}$ and

$\frac{c_1}{c_2}=\frac{1}{4}$

$\therefore$ The condition for the simultaneous equations to have infinitely many solutions is.

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\therefore \frac{1}{4}=\frac{1}{k}=\frac{1}{4}$

$\therefore \frac{1}{4}=\frac{1}{k}$

$\therefore k=4$.

Hence the simultaneous equations $4x+y=7$ and $16x+ky=28$ will have infinitely many solutions for $k=4$.