Solver Solving a System of Linear Equations by Elimination/Addition

Source code of 'Solving a System of Linear Equations by Elimination/Addition'

This Solver (Solving a System of Linear Equations by Elimination/Addition) was created by by jim_thompson5910(35256)

: View Source, Show, Put on YOUR site
About jim_thompson5910: If you need more math help, then you can email me. I charge $2 for steps, or $1 for answers only. Email: jim_thompson5910@hotmail.com Website: http://www.freewebs.com/jimthompson5910/home.html

==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=2]y=*[input c=11]
*[input d=5]x+ *[input e=3]y=*[input f=34]

==section solution
perl

my $check=0, $store=0, $store1=0, $temp=0, $temp1=0, $temp2=0, $count=0, $gcf=1, $lcm=1, @ans1, @ans2, @xans, @yans, $size1, $size2, $xans1, $yans1, $xint, $yint, $inf=0;

sub clean{

@array=();

@hold=@array;

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

for($i=0;$i<3;$i++)
{
$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]=~/\-/))
 {delete $array[$count];
 delete $array[$count+1];
}}

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]=~/\+/))
 {delete $array[$count];}}

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]=~/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]=~/ /))&&($array[$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]=~/\+/))
 {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];
}}
}

#$string=@array;

$string=@hold;

}

##########
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+)/)
  {$num1=$1;
   $denom1=$2;
print "TEST";
   $num1 = -1*$num1 if $a =~ m/\-/;
  }
 else
  {$num1=$a;
   $denom1=1;
   #$num1 = -1*$num1 if $a =~ m/\-/;
  }

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

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

$lcm1=multilcm($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+)/)
  {$num1=$1;
   $denom1=$2;
   $num1 = -1*$num1 if $d =~ m/\-/;
  }
 else
  {$num1=$d;
   $denom1=1;}

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

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

if(($a*$e-$d*$b)!=0)
{$xans1=($c*$e-$f*$b)/($a*$e-$d*$b);

$yans1=($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;
}
}

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{

$temp1=$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;
}
$temp_frac1[0]=$numer1+$numer2;
$temp_frac2[0]=$denom1;
return ($temp_frac1[0],"/",$temp_frac2[0]);
}

if($a!=0)
{$checker=$d/$a;}

if((($a==0)||($b==0))||($c==0))
{
 $check=1;
}

if($check==0)
{
if((($d/$a==$checker)&&($f/$c==$checker))&&($e/$b==$checker))
{

$inf=1;

#return undef;
}

if((($d/$a==$checker)&&($f/$c!=$checker))&&($e/$b==$checker))
{

$inf=2;

#return undef;
}
}

print "

Lets start with the given system of linear equations";

$string="$a*x+$b*y=$c";

$string=clean();

@initial1=@array;

$string="$d*x+$e*y=$f";

$string=clean();

@initial2=@array;

print "

{{{@initial1}}}
{{{@initial2}}}";

for($count=2;$count<abs($a*$d);$count++)
{ if(($a%$count==0)&&($d%$count==0))
   {$gcf=$count;}
}

$lcm=($a*$d)/$gcf;

if($a!=0)
{$store=$lcm/$a;}
if($d!=0)
{$store1=0-$lcm/$d;}

print"

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get $a and $d to some equal number, we could try to get them to the LCM.

Since the LCM of $a and $d is $lcm, we need to multiply both sides of the top equation by $store and multiply both sides of the bottom equation by $store1 like this:";

$string="$store*($a*x+$b*y)=($c)*$store";

$string=clean();

print "

{{{@array}}} Multiply the top equation (both sides) by $store";

$string="$store1*($d*x+$e*y)=($f)*$store1";

$string=clean();

print "
{{{@array}}} Multiply the bottom equation (both sides) by $store1
";

$temp=$store*$a;
$temp1=$store*$b;
$temp2=$store*$c;

$temp3=$store1*$d;
$temp4=$store1*$e;
$temp5=$store1*$f;

print "

So after multiplying we get this:";

$string="$temp*x+$temp1*y=$temp2";

$string=clean();

print "
{{{@array}}}";

$string="$temp3*x+$temp4*y=$temp5";

$string=clean();

if($inf==1)
{
print "
{{{@array}}}

Notice how $temp and $temp3 add to zero, $temp1 and $temp4 add to zero, $temp2 and $temp5 and to zero (ie {{{$temp+$temp3=0}}}) {{{$temp1+$temp4=0}}}, and {{{$temp2+$temp5=0}}})

So we're left with

{{{0=0}}}

which means any x or y value will satisfy the system of equations. So there are an infinite number of solutions

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($inf==2)
{
$temp22=$temp2+$temp5;
print "
{{{@array}}}

Notice how $temp and $temp3 and $temp2 and $temp4 add to zero (ie {{{$temp+$temp3=0}}} {{{$temp1+$temp4=0}}})

However $temp2 and $temp5 add to $temp22 (ie {{{$temp2+$temp5=$temp22}}});

So we're left with

{{{0=$temp22}}}

which means no value of x or y value will satisfy the system of equations. So there are no solutions

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;

}

print "
{{{@array}}}

Notice how $temp and $temp3 add to zero (ie {{{$temp+$temp3=0}}})

Now add the equations together. In order to add 2 equations, group like terms and combine them";

$string="($temp*x+$temp3*x)+($temp1*y+$temp4*y)=$temp2+$temp5";

$string=clean();

print "
{{{@array}}}";

$string="($temp+$temp3)*x+($temp1+$temp4)y=$temp2+$temp5";

$string=clean();

print "

{{{@array}}}

$string="($temp1+$temp4)*y=$temp2+$temp5";

$string=clean();

print "{{{cross($temp+$temp3)*x+@array}}} Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.

$temp1=$temp1+$temp4;
$temp2=$temp2+$temp5;

$string="$temp1*y=$temp2";

$string=clean();

print "
So after adding and canceling out the x terms we're left with:

{{{@array}}}";

$string="$temp2/$temp1";

$string=clean();

print "

{{{y=@array}}} Divide both sides by {{{$temp1}}} to solve for y

$num=$temp2;
$denom=$temp1;
@ans1=reduce();
@yans=@ans1;

$string="@ans1";

$string=clean();

@clean_ans1=@array;

print "

{{{y=@array}}} Reduce

Now plug this answer into the top equation";

$string="$a*x+$b*y=$c";

$string=clean();

print " {{{@array}}} to solve for x";

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

$string=clean();

print "

{{{@array}}} Plug in {{{y=@clean_ans1}}}
";

$ans1[0]=$ans1[0]*$b;

$string="$a*x+@ans1=$c";

$string=clean();

print "

{{{@array}}} Multiply

$size1=@ans1;

if($size1>1)
{$num=$ans1[0];
$denom=$ans1[2];
@ans1=reduce();

$string="@ans1";
$string=clean();

$string="$a*x+@ans1=$c";

$string=clean();

print "

{{{@array}}} Reduce

";}

$string="$a*x=$c-@ans1";

$string=clean();

print "

{{{@array}}} Subtract ";

$string="@ans1";

$string=clean();

print "{{{@array}}} from both sides";

$size1=@ans1;

if($size1>1)
{$temp=$c*$ans1[2];

$string="$a*x=$temp/$ans1[2]-@ans1";

$string=clean();
print "

{{{@array}}} Make $c into a fraction with a denominator of $ans1[2]";}

$size1=@ans1;

if($size1>1)
{$numer1=$c;
$denom1=1;
$numer2=0-$ans1[0];
$denom2=$ans1[2];
@ans1=add();}

if($size1==1)
{@ans1=$c-$ans1[0];}

$string="$a*x=@ans1";

$string=clean();

print "

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

{{{cross((1/$a)($a))*x=(@ans1)(1/$a)}}} Multiply both sides by {{{1/$a}}}. This will cancel out {{{$a}}} on the left side.
";

$size1=@ans1;

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

$num=$ans1[0];
$denom=$ans1[2];
@ans1=reduce();}

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

$num=$ans1[0];
$denom=$ans1[2];
@ans1=reduce();}

$string="@ans1";

$string=clean();

print "

{{{x=@array}}} Multiply the terms on the right side
";

@xans=@ans1;

print "

So our answer is ";

$string="@ans1";

$string=clean();

@x_final=@array;

$string="@yans";

$string=clean();

@y_final=@array;

print "

{{{x=@x_final}}},  {{{y=@y_final}}}

which also looks like

({{{@x_final}}}, {{{@y_final}}})";

$size1=@xans;
$size2=@yans;

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

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

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

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

if($xans1<0)
{$xmin=$xans1-10;
$xmax=(0-$xans1)+10;}

if($xans1>=0)
{$xmax=$xans1+10;
$xmin=(0-$xans1)-10;}

if($yans1<0)
{$ymin=$yans1-10;
$ymax=(0-$yans1)+10;}

if($yans1>=0)
{$ymax=$yans1+10;
$ymin=(0-$yans1)-10;}

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

print "

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( $xint, $yint, $rad ) ) 
)
}}} graph of {{{@initial1}}} (red) {{{@initial2}}} (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).

and we can see that the two equations intersect at ({{{@x_final}}},{{{@y_final}}}). This verifies our answer.";

==section output

==section check