Question

Find those values of c for which the equations:
$2x+3y=3$
$(c+2)x+(c+4)y=c+6$
$(c+2{)}^{2}x+(c+4{)}^{2}y=(c+6{)}^2$
are consistent. Also solve above equations for these values of c.

Answer 1

$2x+3y=3\cdots \left(1\right)$

$(c+2)x+(c+4)y=c+6\cdots \left(2\right)$

$(c+2{)}^{2}x+(c+4{)}^{2}y=(c+6{)}^2\cdots (3)$

solving $\left(1\right)$ and $\left(2\right)$

we get $x=-\frac{c+6}{c+2}$ and $y=2(\frac{c+6}{c+4})$

for equations to be consistent, $x,y$ must satisfy equation $\left(1\right)$

$⟹-2(\frac{c+6}{c+2})+6(\frac{c+6}{c+4})=3$

solving this we get $c^2+10c-12=0⟹c=-5\pm \sqrt{37}$

$\therefore x=\frac{-115\mp \sqrt{37}}{28},y=\frac{73\pm \sqrt{37}}{9}$