Solver Solving a linear system of equations by subsitution

Source code of 'Solving a linear system of equations by subsitution'

This Solver (Solving a linear system of equations by subsitution) was created by by jim_thompson5910(35256)

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==section input
Enter the system of equations. Make sure every coefficient is a rational number. For instance if you have {{{0.2x+0.25y=10}}} enter the expression {{{(1/5)x+(1/4)y=10}}}

*[input a=1]x+ *[input b=3]y=*[input c=4]
*[input d=4]x+ *[input e=6]y=*[input f=8]

==section solution
perl

my $check=0, $store=1, $store1=1, $store2=1, $store3=1, $store5=1, $temp1, $temp2, $numer1, $denom1, $numer2, $denom2, @temp_frac1, @temp_frac2, $count=2, $size1, $size2, $inf=0, $xans, $yans;

my $string1, $string2, $num=1, $denom=1, $quotient, $i=1, $j=1, @frac1, @frac2, @ans1, @ans2, $gcf, $lcm, $xmin, $xmax, $ymin, $ymax, $xint, $yint;

sub clean{

@array=();

@array=split(//,$string);

$size=@array;

for($count=0;$count<$size;$count++)
{if(($array[$count]=~/\+/)&&($array[$count+1]=~/\-/))
 { delete $array[$count];
}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\+/)&&($array[$count+1]=~/\(/))&&($array[$count+2]=~/\-/))
 {$array[$count]="-";
 delete $array[$count+2];
}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\-/)&&($array[$count+1]=~/\(/))&&($array[$count+2]=~/\-/))
 {$array[$count]="+";
 delete $array[$count+2];
}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\-/)&&($array[$count+1]=~/0/))&&($array[$count+2]!~/\d/))
 {$array[$count]="+";
}}

for($count=0;$count<$size;$count++)
{if(($array[$count]=~/\-/)&&($array[$count+1]=~/\+/))
 { delete $array[$count+1];
}}

for($count=0;$count<$size;$count++)
{if(($array[$count]=~/\-/)&&($array[$count+1]=~/\-/))
 {delete $array[$count];
  $array[$count+1]="+";}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\//)&&($array[$count+1]=~/1/))&&(($array[$count+2]=~/\)/)&&($array[$count+3]=~/\*/)))
 {delete $array[$count];
 delete $array[$count+1];
delete $array[$count+2];
   for($count1=$count+2;$count1>=0;$count1--)
     {if($array[$count1]=~/\(/)
        {delete $array[$count1];
          $count1=-1;}
     }
}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\//)&&($array[$count+1]=~/1/))&&($array[$count+2]=~/\+/))
 {delete $array[$count];
 delete $array[$count+1];
}}

for($count=0;$count<$size;$count++)
{if((($array[$count]=~/\//)&&($array[$count+1]=~/1/))&&($array[$count+2]!~/\d/))
 {delete $array[$count];
 delete $array[$count+1];
}}

}

##########
sub gcf {
  my ($x, $y) = @_;
  ($x, $y) = ($y, $x % $y) while $y;
  return $x;
}

sub lcm {
  return($_[0] * $_[1] / gcf($_[0], $_[1]));
}

sub multigcf {
  my $x = shift;
  $x = gcf($x, shift) while @_;
  return $x;
}

sub multilcm {
  my $x = shift;
  $x = lcm($x, shift) while @_;
  return $x;
}

##################

## Fraction support

if((($a=~m/\//)||($b=~m/\//))||($c=~m/\//))
{
 if(($a=~m/(\-\d+)\/(\d+)/)||($a=~m/(\d+)\/(\d+)/))
  {$num1=$1;
   $denom1=$2;}
 else
  {$num1=$a;
   $denom1=1;}

if(($b=~m/(\-\d+)\/(\d+)/)||($b=~m/(\d+)\/(\d+)/))
  {$num2=$1;
   $denom2=$2;}
 else
  {$num2=$b;
   $denom2=1;}

if(($c=~m/(\-\d+)\/(\d+)/)||($c=~m/(\d+)\/(\d+)/))
  {$num3=$1;
   $denom3=$2;}
 else
  {$num3=$c;
   $denom3=1;}

$lcm1=multilcm($denom1,$denom2,$denom3);

print "

{{{($a)*x+($b)*y=$c}}} Start with the first equation

{{{$lcm1(($a)*x+($b)*y)=($lcm1)*($c)}}} Multiply both sides by the LCD $lcm1

$a=$lcm1*($num1/$denom1);
$b=$lcm1*($num2/$denom2);
$c=$lcm1*($num3/$denom3);

print "

{{{$a*x+$b*y=$c}}}Distribute and simplify

------------------------------------------

";
}

if((($d=~m/\//)||($e=~m/\//))||($f=~m/\//))
{
 if(($d=~m/(\-\d+)\/(\d+)/)||($d=~m/(\d+)\/(\d+)/))
  {$num1=$1;
   $denom1=$2;}
 else
  {$num1=$d;
   $denom1=1;}

if(($e=~m/(\-\d+)\/(\d+)/)||($e=~m/(\d+)\/(\d+)/))
  {$num2=$1;
   $denom2=$2;}
 else
  {$num2=$e;
   $denom2=1;}

if(($f=~m/(\-\d+)\/(\d+)/)||($f=~m/(\d+)\/(\d+)/))
  {$num3=$1;
   $denom3=$2;}
 else
  {$num3=$f;
   $denom3=1;}

$lcm1=multilcm($denom1,$denom2,$denom3);

print "

{{{($d)*x+($e)*y=$f}}} Start with the second equation

{{{$lcm1(($d)*x+($e)*y)=($lcm1)*($f)}}} Multiply both sides by the LCD $lcm1

$d=$lcm1*($num1/$denom1);
$e=$lcm1*($num2/$denom2);
$f=$lcm1*($num3/$denom3);

print "

{{{$d*x+$e*y=$f}}} Distribute and simplify

-----------------------------------------

";
}

## END Fraction support

sub invert{
$num=$denom1;
$denom=$numer1;
return ($num,"/", $denom);
}

if(($a*$e-$d*$b)!=0)
{$xans=($c*$e-$f*$b)/($a*$e-$d*$b);

$yans=($a*$f-$d*$c)/($a*$e-$d*$b);}

sub reduce{

if($denom!=0)
{
$store=$num/$denom;

if($store>0)
{
  $num=abs($num);
  $denom=abs($denom);
}

if(($store<0)&&($denom<0))
{
  $num=0-$num;
  $denom=abs($denom);
}

if($num%$denom==0)
{
  $quotient=$num/$denom;
  return ($quotient,"/",1);
}
}

for ($i=abs($num*$denom); $i>1; $i--)
{
 if (($num%$i==0)&&($denom%$i==0))
    { 
       $num/=$i;
       $denom/=$i;
       $j=$i;
      
    }
}
return ($num,"/",$denom);
}

sub mult{

$string1="@frac1";
$string2="@frac2";

$temp1=$numer1*$numer2;
$temp2=$denom1*$denom2;
return ($temp1,"/",$temp2);
}

sub add{

if(($denom1!=$denom2)&&(($denom1!=0)&&($denom2!=0)))
{
 for($count=1;$count<abs($denom1*$denom2);$count++)
  {if(($denom1%$count==0)&&($denom2%$count==0))
    {$gcf=$count;}
  }
$lcm=($denom1*$denom2)/$gcf;
$numer1=($lcm/$denom1)*$numer1;
$numer2=($lcm/$denom2)*$numer2;

$denom1=($lcm/$denom1)*$denom1;
$denom2=($lcm/$denom2)*$denom2;
}
$tempfrac1[0]=$numer1+$numer2;
$tempfrac2[0]=$denom1;
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub subtract{

$denom1=($lcm/$denom1)*$denom1;
$denom2=($lcm/$denom2)*$denom2;
}
$temp1=$numer1-$numer2;
$temp2=$denom1;
return ($temp1,"/",$temp2);
}

if($check==0)
{
if(($b!=0)&&($c!=0))
{
if((($d/$a==$e/$b)&&($f/$c==$e/$b))&&($d/$a==$f/$c))
{$inf=1;}
}
}

$string="$a*x+$b*y=$c";
$string=clean();
@initial1=@array;

$string="$d*x+$e*y=$f";
$string=clean();
@initial2=@array;

print "

Lets start with the given system of linear equations

{{{@initial1}}}
{{{@initial2}}}

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.";

$string="$b*y=$c-$a*x";
$string=clean();

$abs=abs($a);

print "

Solve for y for the first equation

{{{@array}}}", ($a>0)? "Subtract {{{$abs*x}}} from ":"Add {{{$abs*x}}} to ";

print "both sides";

$string="y=($c-$a*x)/$b";
$string=clean();

print "

{{{@array}}} Divide both sides by $b.

Which breaks down and reduces to

$num=$c;
$denom=$b;
@frac1=reduce();
$num=0-$a;
$denom=$b;
@frac2=reduce();

$string="y=@frac1+(@frac2)*x";
$string=clean();

print "

{{{@array}}} Now we've fully isolated y";

$string="@frac1+(@frac2)*x";
$string=clean();
@temp_array=@array;

print "

Since y equals {{{@array}}} we can substitute the expression {{{@array}}} into y of the 2nd equation. This will eliminate y so we can solve for x.

{{{$d*x+$e*highlight((@array))=$f}}} Replace y with {{{@array}}}. Since this eliminates y, we can now solve for x.";

$string="$d*x+$e*(@frac1)+$e(@frac2)x=$f";
$string=clean();

print "

{{{@array}}} Distribute $e to {{{@temp_array}}}";

$frac1[0]=$e*$frac1[0];
$frac2[0]=$e*$frac2[0];

$string="$d*x+@frac1+(@frac2)*x=$f";
$string=clean();

print "

{{{@array}}} Multiply

";
$reduce=0;

$size1=@frac1;
$size2=@frac2;

if($size1>1)
{$num=$frac1[0];
$denom=$frac1[2];
@frac1=reduce();
$reduce++;}

if($size1==1)
{$num=$frac1[0];
$denom=1;
@frac1=reduce();}

if($size2>1)
{$num=$frac2[0];
$denom=$frac2[2];
@frac2=reduce();
$reduce++;}

if($size2==1)
{$num=$frac2[0];
$denom=1;
@frac2=reduce();}

if($reduce>0)
{$string="$d*x+@frac1+(@frac2)*x=$f";
$string=clean();

print "

{{{@array}}} Reduce any fractions";}

@abs=@frac1;
$abs[0]=abs($abs[0]);

$string="@abs";
$string=clean();
@abs=@array;

$string="$d*x+(@frac2)*x=$f-@frac1";
$string=clean();

print "

{{{@array}}}", ($frac1[0]>0) ? " Subtract {{{@abs}}} from both sides" : "Add {{{@abs}}} to both sides";

$size1=@frac1;

if($frac1[2]!=1)
{$store=$frac1[2];
$temp1=$f*$store;

$string="$d*x+(@frac2)*x=$temp1/$store-@frac1";
$string=clean();

print "

{{{@array}}} Make $f into a fraction with a denominator of $store";
}

if($frac1[2]!=1)
{
$numer1=$f;
$denom1=1;
$numer2=$frac1[0];
$denom2=$frac1[2];
@temp_frac1=subtract();}

if($frac1[2]==1)
{@temp_frac1=($f-$frac1[0],"/","1");}

$string="$d*x+(@frac2)*x=@temp_frac1";
$string=clean();

print "

{{{@array}}} Combine the terms on the right side

$size2=@frac2;
if($frac2[2]!=1)
{
$store=$d;
$store=$frac2[2]*$store;

$string="($store/$frac2[2])*x+(@frac2)x=@temp_frac1";
$string=clean();

print "

{{{@array}}} Make $d into a fraction with a denominator of $frac2[2]";

$numer1=$d;
$denom1=1;
$numer2=$frac2[0];
$denom2=$frac2[2];
@temp_frac2=add();

}

if($frac2[2]==1)
{@temp_frac2=($d+$frac2[0],"/","1");}

$string="(@temp_frac2)*x=@temp_frac1";
$string=clean();

print "

{{{@array}}} Now combine the terms on the left side.  ";

if(($temp_frac2[0]==0)&&($inf==1))
{
$string="@temp_frac2=@temp_frac1";
$string=clean();

print "
 {{{@array}}} Since this expression is true for any x, we have an identity.

So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph {{{@initial1}}} and {{{@initial2}}} we get

{{{ graph( 500, 600, -6, 5, -10, 10, ($c-$a*x)/$b) }}} graph of {{{@initial1}}}

{{{ graph( 500, 600, -6, 5, -10, 10, ($f-$d*x)/$e ) }}} graph of {{{@initial2}}} (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is <a href=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>dependent</a>";

return undef;
}

if($temp_frac2[0]==0)
{
print "
 {{{@temp_frac2=@temp_frac1}}} Since this expression is not true, we have an inconsistency.

So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.

{{{ graph( 500, 600, -6, 5, -10, 10, ($c-$a*x)/$b, ($f-$d*x)/$e ) }}} graph of {{{@initial1}}} (red) and {{{@initial2}}} (green) (hint: you may have to solve for y to graph these)

and we can see that the two equations are parallel and will never intersect. So this system is <a href=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>inconsistent</a>";

return undef;
}

$size1=@temp_frac2;

if($size1>1)
{$numer1=$temp_frac2[0];
$denom1=$temp_frac2[2];
@frac1=invert();}

if($size1==1)
{$numer1=$temp_frac2[0];
$denom1=1;
@frac1=invert();}

print "

{{{cross((@frac1)(@temp_frac2))x=(@temp_frac1)(@frac1)}}} Multiply both sides by {{{@frac1}}}. This will cancel out {{{@temp_frac2}}} and isolate x

So when we multiply {{{@temp_frac1}}} and {{{@frac1}}} (and simplify) we get

$size1=@temp_frac1;

if($size1>1)
{
$numer1=$temp_frac1[0];
$denom1=$temp_frac1[2];
$numer2=$frac1[0];
$denom2=$frac1[2];
@temp_frac1=mult();

$num=$temp_frac1[0];
$denom=$temp_frac1[2];
@temp_frac1=reduce();

}

if($size1==1)
{
$numer1=$temp_frac1[0];
$denom1=1;
$numer2=$frac1[0];
$denom2=$frac1[2];
@temp_frac1=mult();

$num=$temp_frac1[0];
$denom=$temp_frac1[2];
@temp_frac1=reduce();

}

if((($temp_frac1[0]<0)||($temp_frac1[2]<0))&&(@temp_frac1==0))
{$temp_frac1[0]=abs($temp_frac1[0]);
 $temp_frac1[2]=abs($temp_frac1[2]);}

$string="@temp_frac1";
$string=clean();
@clean_ans1=@array;

print "

{{{x=@clean_ans1}}} <---------------------------------One answer

@ans1=@temp_frac1;

print " Now that we know that {{{x=@clean_ans1}}}, lets substitute that in for x to solve for y";

$string="$d(@ans1)+$e*y=$f";
$string=clean();

print "

{{{@array}}} Plug in {{{x=@clean_ans1}}} into the 2nd equation ";

$size1=@ans1;

if($size1>1)
{
$numer1=$d;
$denom1=1;
$numer2=$ans1[0];
$denom2=$ans1[2];
@temp_frac1=mult();

$num=$temp_frac1[0];
$denom=$temp_frac1[2];
@temp_frac1=reduce();

}

if($size1==1)
{
$numer1=$d;
$denom1=1;
$numer2=$ans1[0];
$denom2=1;
@temp_frac1=mult();

$num=$temp_frac1[0];
$denom=$temp_frac1[2];
@temp_frac1=reduce();

}

$string="@temp_frac1+$e*y=$f";
$string=clean();

print "

{{{@array}}} Multiply";

@abs=@temp_frac1;
$abs[0]=abs($abs[0]);

$string="@abs";
$string=clean();
@abs=@array;

$string="$e*y=$f-@temp_frac1";
$string=clean();

print "

{{{@array}}}", ($temp_frac1[0]>0) ? "Subtract {{{@abs}}} from both sides" :  "Add {{{@abs}}} to both sides";

$size1=@temp_frac1;

if($temp_frac1[2]==1)
{$numer1=$f;
$denom1=1;
$numer2=$temp_frac1[0];
$denom2=1;
@temp_frac1=subtract();}

if($temp_frac1[2]!=1)
{$store=$f;
$temp1=$store*$temp_frac1[2];

$string="$e*y=$temp1/$temp_frac1[2]-@temp_frac1";
$string=clean();

print "

{{{@array}}} Make $f into a fraction with a denominator of $temp_frac1[2]

$numer1=$f;
$denom1=1;
$numer2=$temp_frac1[0];
$denom2=$temp_frac1[2];
@temp_frac1=subtract();}

$string="$e*y=@temp_frac1";
$string=clean();

#{{{$e*y=$f-@temp_frac1}}}

print "

{{{@array}}} Combine the terms on the right side ";

$numer1=$e;
$denom1=1;
@temp_frac2=invert();

print "

{{{cross((@temp_frac2)($e))*y=(@temp_frac1)(@temp_frac2)}}} Multiply both sides by {{{@temp_frac2}}}. This will cancel out $e on the left side.";

$size1=@temp_frac1;
$size2=@temp_frac2;

if(($size1>1)&&($size2>1))
{
$numer1=$temp_frac1[0];
$denom1=$temp_frac1[2];
$numer2=$temp_frac2[0];
$denom2=$temp_frac2[2];
@temp_frac1=mult();
}

if(($size1==1)&&($size2>1))
{
$numer1=$temp_frac1[0];
$denom1=1;
$numer2=$temp_frac2[0];
$denom2=$temp_frac2[2];
@temp_frac1=mult();
}

if(($size1>1)&&($size2==1))
{
$numer1=$temp_frac1[0];
$denom1=$temp_frac1[2];
$numer2=$temp_frac2[0];
$denom2=1;
@temp_frac1=mult();
}

if(($size1==1)&&($size2==1))
{
$numer1=$temp_frac1[0];
$denom1=1;
$numer2=$temp_frac2[0];
$denom2=1;
@temp_frac1=mult();
}

$string="y=@temp_frac1";
$string=clean();

print "

{{{y=@temp_frac1}}} Multiply the terms on the right side";

$num=$temp_frac1[0];
$denom=$temp_frac1[2];
@ans2=reduce();

$size1=@ans1;
$size2=@ans2;

if($size1==1)
{
 $xmin=$xans-10;
 $xmax=$xans+10;
}

if($size2==1)
{
 $ymin=$ans2[0]-10;
 $ymax=$ans2[2]+10;
}

if($size1>1)
{$temp1=$ans1[0]/$ans1[2];

$xmin=$temp1-10;
$xmax=$temp1+10; 
}

if($size2>1)
{$temp2=$ans2[0]/$ans2[2];
$ymin=$temp2-10;
$ymax=$temp2+10;
}

$size1=@ans1;
$size2=@ans2;

if($size1>1)
{$xans=$ans1[0]/$ans1[2];}

if($size1==1)
{$xans=$ans1[0];}

if($size2>1)
{$yans=$ans2[0]/$ans2[2];}

if($size2==1)
{$yans=$ans2[0];}

if($xans<0)
{$xmin=$xans-10;
$xmax=(0-$xans)+10;}

if($xans>0)
{$xmax=$xans+10;
$xmin=(0-$xans)-10;}

if($yans<0)
{$ymin=$yans-10;
$ymax=(0-$yans)+10;}

if($yans>0)
{$ymax=$yans+10;
$ymin=(0-$yans)-10;}

$store=abs($xmax-$xmin);
$store1=($ymax-$ymin);
$rad=$store/150;

$string="@ans2";
$string=clean();
@clean_ans2=@array;

$xans1=$ans1[0]/$ans1[2];

$yans1=$ans2[0]/$ans2[2];

print "

{{{y=@clean_ans2}}} Reduce

So this is the other answer

{{{y=@clean_ans2}}}<---------------------------------Other answer

So our solution is

{{{x=@clean_ans1}}} and {{{y=@clean_ans2}}}

which can also look like

({{{@clean_ans1}}},{{{@clean_ans2}}})

Notice if we graph the equations (if you need help with graphing, check out this <a href=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)

{{{@initial1}}}
{{{@initial2}}}

we get

{{{
drawing( 500, 600, $xmin, $xmax, $ymin, $ymax,
  graph( 500, 600, $xmin, $xmax, $ymin, $ymax, ($c-$a*x)/$b, ($f-$d*x)/$e ),
  blue(  circle( $xans1, $yans1, $rad ) ) 
)
}}}graph of {{{@initial1}}} (red) and {{{@initial2}}} (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.

and we can see that the two equations intersect at ({{{@clean_ans1}}},{{{@clean_ans2}}}). This verifies our answer.

-----------------------------------------------------------------------------------------------
Check:

Plug in ({{{@clean_ans1}}},{{{@clean_ans2}}}) into the system of equations

Let {{{x=@clean_ans1}}} and {{{y=@clean_ans2}}}. Now plug those values into the equation {{{@initial1}}}";

$string="$a*(@ans1)+$b*(@ans2)=$c";
$string=clean();

print "

{{{@array}}} Plug in {{{x=@clean_ans1}}} and {{{y=@clean_ans2}}}
";

$numer1=$a;
$denom1=1;
$numer2=$ans1[0];
$denom2=$ans1[2];
@storage1=mult();

$numer1=$b;
$denom1=1;
$numer2=$ans2[0];
$denom2=$ans2[2];
@storage2=mult();

$string="@storage1+@storage2=$c";
$string=clean();

print "

{{{@array}}} Multiply
";

$numer1=$storage1[0];
$denom1=$storage1[2];
$numer2=$storage2[0];
$denom2=$storage2[2];
@storage1=add();

$string="@storage1=$c";
$string=clean();

print "

{{{@array}}} Add
";

$num=$storage1[0];
$denom=$storage1[2];
@storage1=reduce();

$string="@storage1=$c";
$string=clean();

print "

{{{@array}}} Reduce. Since this equation is true the solution works.
";

print "

So the solution ({{{@clean_ans1}}},{{{@clean_ans2}}}) satisfies {{{@initial1}}}

print "

Let {{{x=@clean_ans1}}} and {{{y=@clean_ans2}}}. Now plug those values into the equation {{{@initial2}}}";

$string="$d*(@ans1)+$e*(@ans2)=$f";
$string=clean();

print "

{{{@array}}} Plug in {{{x=@clean_ans1}}} and {{{y=@clean_ans2}}}
";

$numer1=$d;
$denom1=1;
$numer2=$ans1[0];
$denom2=$ans1[2];
@storage1=mult();

$numer1=$e;
$denom1=1;
$numer2=$ans2[0];
$denom2=$ans2[2];
@storage2=mult();

$string="@storage1+@storage2=$f";
$string=clean();

print "

{{{@array}}} Multiply
";

$numer1=$storage1[0];
$denom1=$storage1[2];
$numer2=$storage2[0];
$denom2=$storage2[2];
@storage1=add();

$string="@storage1=$f";
$string=clean();

print "

{{{@array}}} Add
";

$num=$storage1[0];
$denom=$storage1[2];
@storage1=reduce();

$string="@storage1=$f";
$string=clean();

print "

{{{@array}}} Reduce. Since this equation is true the solution works.
";

print "

So the solution ({{{@clean_ans1}}},{{{@clean_ans2}}}) satisfies {{{@initial2}}}

Since the solution ({{{@clean_ans1}}},{{{@clean_ans2}}}) satisfies the system of equations

{{{@initial1}}}
{{{@initial2}}}

this verifies our answer.

==section output

==section check
  angle=40 angle1=50