Question

Prove that there is a value of c for which the system $cx+2y=c-2,8x+cy=c$ has infinitely many solutions. Find this value.

Answer 1

Two equations given are

$cx+2y=c-2$

$8x+cy=c$

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

$\frac{c}{8}=\frac{2}{c}=\frac{c-2}{c}$

Now $\frac{c}{8}=\frac{2}{c}$

$c^2=16$

$c=\pm 4$

now $\frac{2}{c}=\frac{c-2}{c}$

$c^2-4c=0$

$c(c-4)=0$

$c=0,4$

so only common value of c is $4$

so for $c=4$ this system equations posses the infinite solutions.