Solving the second example problem from the Wikipedia page on evelopes

Question

While reading the Wikipedia page on Envelopes there are some examples given. In the second example a jump is made from a linear equation to $F(x,y,t)=0$ form. For context the transformation is made from the linear equation of $y=\frac{-(k-t)x}{t}+k-t$ to $F(x,y,t)=t^2+t(y-x-k)+kx=0$, where. $k$ is an arbitrary scaling variable and $t$ is a parameter use to produce the family of lines. I was wondering if someone could explain to me home this transition is made? If you would like to the at the original page here is a link.

Answer 1

Multiply by $t$ and bring all to one side