how do I get the following plane equation and the result t from it? steps

Question

Below are ray equations substituting in the plane equation. How did I get to rearranging t to isolating t?

given rays of the form:

$x = a(t)+a0$

$y = b(t) + b0$

$z = c(t) + c0$

given plane for the form:

$Ax * By * Cz + D = 0$

Substituting the ray into the plane: $ A(at + a0)+B(bt + b0)+C(ct + c0) + D = 0 $

The following is "t". My question is how do we go from the equation above(Substituting the ray into the plane: ) to isolating t? I need steps.

$ t = (D+(a0A)+(b0B)+(c0C))/(cC+bB+aA) $

Answer 1

Distributing $A(at + a_0)+B(bt + b_0)+C(ct + c_0) + D = 0$, we get: $Aat + Aa_0 + Bbt + Bb_0 + Cct + Cc_0 + D = 0$.

Factorize: $(Aa + Bb + Cc)t + Aa_0 + Bb_0 + Cc_0 + D = 0$.

If this bad boy $Aa + Bb + Cc$ isn't zero, we have: $t = -(Aa_0 + Bb_0 + Cc_0 + D)/(Aa + Bb + Cc)$.