Full Code

unit Unit1;

{$mode objfpc}{$H+}

interface

uses
  Classes, SysUtils, FileUtil, Forms, Controls, Graphics, Dialogs, ComCtrls,
  StdCtrls, ExtCtrls, Buttons, Math;

type

  { TForm1 }

  TForm1 = class(TForm)
    Edit1: TEdit;
    Edit2: TEdit;
    Edit3: TEdit;
    Edit4: TEdit;
    Edit5: TEdit;
    Edit6: TEdit;
    Edit7: TEdit;
    Edit8: TEdit;
    Image1: TImage;
    Label1: TLabel;
    Label2: TLabel;
    Label3: TLabel;
    Label4: TLabel;
    Label5: TLabel;
    PageControl1: TPageControl;
    SpeedButton1: TSpeedButton;
    StatusBar1: TStatusBar;
    TabSheet1: TTabSheet;
    TabSheet2: TTabSheet;
    procedure SpeedButton1Click(Sender: TObject);
  private
    { private declarations }
  public
    { public declarations }
  end;

var
  Form1: TForm1;
  x,y,detx,dety,det: double;

implementation

{$R *.lfm}

{ TForm1 }

function getFrazione(x: double):string;
var h1, h2, k1, k2, y, a, aux: double;
begin

  //Assegnazione di valori
  h1 := 1;
  h2 := 0;
  k1 := 0;
  k2 := 1;
  y := x;

  repeat
   begin
    a := floor(y);
    aux := h1;
    h1 := a*h1+h2;
    h2 := aux;
    aux := k1;
    k1 := a*k1+k2;
    k2 := aux;
    y := 1/(y-a) ;
   end;
   until ( abs(x-h1/k1) > x*0.000001);

  Result := FloatToStr(h1) + '/' + FloatToStr(k1);
end;

procedure TForm1.SpeedButton1Click(Sender: TObject);
var a,b,c,d,e,f: double;
begin

  a := StrToFloat(Edit1.text);
  b := StrToFloat(Edit2.text);
  c := StrToFloat(Edit3.text);
  d := StrToFloat(Edit4.text);
  e := StrToFloat(Edit5.text);
  f := StrToFloat(Edit6.text);

  {calcolo nella matrice}
  detx := (c*e)-(b*f);
  dety := (a*f)-(c*d);
  det  := (a*e)-(d*b);

  {esportazione dati}
  if ( det <> 0 ) then
  begin
    x := detx/det;
    y := dety/det;
    Edit7.Text := getFrazione(x);
    Edit8.Text := getFrazione(y);
  end
   else
    if ((detx=0) and (dety=0)) then
     begin
      Edit7.Clear;
      Edit8.Clear;
     end
    else
   begin
    Edit7.Clear;
    Edit8.Clear;
  end;
end;

end.