I'm stuck and don't know what to do next to solve this equation for $y$ in terms of $x$. Any hints?
$$ y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1) $$
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$\begingroup$Your equation is: $$y(x_2-x_1)-y_1(x_2-x_1)=x(y_2-y_1)-x_1(y_2-y1)$$ Group everything on the left side of the equation: $$y(x_2−x_1)-x(y_2−y_1)+x_1(y_2−y_1)-y_1(x_2−x_1)=0$$
As I assume you know $x_1$ $x_2$ $y_1$ and $y_2$, it yields an equation like this: $$ax+by+c=0$$ with $a = y_1-y_2$; $b = x_2-x_1$ and $c=x_1y_2-y_1x_2$ which are known values.
You then have a affine line of solutions if $a \ne 0$ and $b \ne 0$: $$x = -\dfrac{by+c}{a}$$ if $b = 0$ and $a \ne 0$, the former equation stands, but can be rewritten: $y \in \mathbb{R}$ and $x = -\dfrac{c}{a}$
if $a = 0$ and $b \ne 0$ you have another affine line of solutions: $x \in \mathbb{R}$ and $y = -\dfrac{b}{c}$.
Finally if $a = b = 0$, there is a solution iff $c = 0$, and this solutions will be all $x,y \in \mathbb{R}²$
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