Question

For what values of $a$ does each of the following system of equations have an infinite number of solutions?
$(a-2)x+27y=4.5,2x+(a+1)y=-1$

Answer 1

Given equations are

$(a-2)x+27y=4.5$

$2x+(a+1)y=-1$

For a system of equations $a_{1}x+b_{1}y=c_1$

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

To have infinite solutions,

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

$⟹\frac{a-2}{2}=\frac{27}{a+1}=\frac{4.5}{-1}$

$⟹(a-2)=-9$

$⟹a=-7$

$27=\frac{45}{10}(-1)(a+1)$

$a+1+6=0$

$⟹a=-7$