Solver SOLVE linear system by SUBSTITUTION

Source code of 'SOLVE linear system by SUBSTITUTION'

This Solver (SOLVE linear system by SUBSTITUTION) was created by by ichudov(507)

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==section input
Variable names: *[input X=x] *[input Y=y]

*[input a11=1]X + *[input a12=-1]Y = *[input b1=0]

*[input a21=1]X + *[input a22=1]Y =  *[input b2=2]

==section solution
perl

print "Solve:
{{{ system( 
  $a11\\$X + $a12\\$Y = $b1,
  $a21\\$X + $a22\\$Y = $b2 )
}}}";

if( $a11 != 0 ) {
  print "We'll use substitution. After moving $a12*$Y to the right, we get:
{{{$a11*$X = $b1 - $a12*$Y}}}, or {{{$X = $b1/$a11 - $a12*$Y/$a11}}}. Substitute that
into another equation:
{{{$a21*($b1/$a11 - $a12*$Y/$a11) + $a22\\$Y = $b2}}} and simplify: ";

my $simplified = solve( "explain_simplification", "$a21*($b1/$a11 - $a12*$Y/$a11) + $a22\\$Y = $b2" );

$y = $simplified;
$y =~ s/^.*?\;\w+=//;
$x = $b1/$a11 - $a12*$y/$a11;
print "So, we know that $Y=$y. Since {{{$X = $b1/$a11 - $a12*$Y/$a11}}}, $X=$x.

Answer: {{{system( $X=$x, $Y=$y )}}}.
";

} else {
  if( $a12 == 0 ) {
    print "You have a bad equation system, in the first equation, both coefficients are 0.";
    return 0;
  }
  print "The first equation is an easy equation for $Y: {{{$Y = $b1/$a12}}}.\n\n";
  $y = $b1/$a12;
  print "Knowing $Y, we can substitute it into the second equation: {{{$a21*$X+$a22*$b1/$a12 = $b2}}}. From here, {{{$X=($b2-$a22*$b1/$a12)/$a21}}}";
  $x = solve( "explain_simplification", "($b2-$a22*$b1/$a12)/$a21" );
  $x =~ s/;//;
  
  print "Your answer: {{{system( $X=$x, $Y=$y ) }}}.\n";
}

==section output
x
y
==section check
 X=x Y=y a11=0 a12=-1 b1=0 a21=1 a22=1 b2=2 x=2 y=-0

X=x Y=y a11=1 a12=-1 b1=0 a21=1 a22=1 b2=2 x=1 y=1

==section practice
perl
  $a11 = randint( 1, 3 );
  $a12 = randint( 1, 3 );
  $a21 = randint( 1, 3 );
  $a22 = randint( 1, 3 );
  my $x1 = randint( -1, 4 );
  my $x2 = randint( -1, 4 );
  $b1 = $a11*$x1 + $a12*$x2;
  $b2 = $a21*$x1 + $a22*$x2;