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I want to reduce this equation to the form of y == kx + b, but I cann't do that using MMA.

Collect[(Det[( {
       {1, x, y},
       {1, a, b},
       {1, c, d}
      } )] == 0) // FullSimplify(*Two points determine a straight line*), {x, y}]
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2 Answers 2

Reset to default
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Try

Solve[Det[({{1, x, y}, {1, a, b}, {1, c, d}})] == 0, y][[1]] //Collect[#, x, Simplify] &
(*{y -> (-b c + a d)/(a - c) + ((b - d) x)/(a - c)}*)

create equation

(% /. Rule -> Equal) [[1]]
(*y == (-b c + a d)/(a - c) + ((b - d) x)/(a - c)*)
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  • $\begingroup$ But it's not in the format of y = k * x + B, it needs some other operations, such as ApplySides. $\endgroup$
    – A little mouse on the pampas
    Commented Jan 16, 2020 at 10:20
  • 1
    $\begingroup$ I modiefied my answer... $\endgroup$
    – Ulrich Neumann
    Commented Jan 16, 2020 at 10:22
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I solve this problem like this, but it's too complicated.

SolveAlways[
 Det[({{1, x, y}, {1, a, b}, {1, c, d}})] == A*y + B*x + CC && 
  k != 0, {x, y}]
Defer[(y == -(B/A)*x - CC/A == 0)] /. %
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