Solver Finding the Equation of a Line Parallel or Perpendicular to a Given Line

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Source code of 'Finding the Equation of a Line Parallel or Perpendicular to a Given Line'

This Solver (Finding the Equation of a Line Parallel or Perpendicular to a Given Line) was created by by jim_thompson5910(13794)

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==section input
Find the equation of a line that is *[choice form parallel perpendicular] to y=*[input m=2]x+*[input b=5] and goes through the point (*[input x1=2],*[input y1=2]).

note: if you have any decimal numbers or coefficients, enter them as fractions. For instance, if you have something like "Find the equation of a line that is parallel to {{{y=0.5x+0.1}}} and goes through (0.2,0.3)", enter it as  "Find the equation of a line that is parallel to {{{y=(1/2)x+1/10}}} and goes through ({{{1/5}}},{{{3/10}}})"

==section solution

perl

if($m=~/\//)
{@m=split(/\//,$m);

$m[2]=$m[1];
$m[1]="/";}

if($m!~/\//)
{$m[0]=$m;
$m[1]="/";
$m[2]=1;}

if($x1=~/\//)
{@x1=split(/\//,$x1);

$x1[2]=$x1[1];
$x1[1]="/";}

if($x1!~/\//)
{$x1[0]=$x1;
$x1[1]="/";
$x1[2]=1;}

if($y1=~/\//)
{@y1=split(/\//,$y1);

$y1[2]=$y1[1];
$y1[1]="/";}

if($y1!~/\//)
{$y1[0]=$y1;
$y1[1]="/";
$y1[2]=1;}

$m1=$m[0]/$m[2];

$xcoord=$x1[0]/$x1[2];
$ycoord=$y1[0]/$y1[2];

if($xcoord==0)
{$xmax=$xcoord+5;
$xmin=0-5;}

if($xcoord>0)
{$xmax=$xcoord+5;
$xmin=0-5;}

if($xcoord<0)
{$xmax=5;
$xmin=$xcoord-5;}

if($ycoord>$b)
{$ymax=$ycoord+5;
$ymin=$b-5;}

if($b>$ycoord)
{$ymax=$b+5;
$ymin=$ycoord-5;}

if($b==$ycoord)
{$ymax=$b+5;
$ymin=$ycoord-5;}

if((($x1[0]==0)&&($y1[0]==$b))&&($form  eq 'parallel'))
{$string="y=(@m)*x+$b";
$string=clean();
@clean_start=@array;

$string="@x1";
$string=clean();
@cleanx=@array;

$string="@y1";
$string=clean();
@cleany=@array;

print "The equation {{{@clean_start}}} goes through the point ({{{@cleanx}}},{{{@cleany}}}) (which is equal and parallel to the given equation)";
return undef;}

sub reduce{

if($arr1[2]!=0)
{
$store=$arr1[0]/$arr1[2];

if($store>0)
{
  $arr1[0]=abs($arr1[0]);
  $arr1[2]=abs($arr1[2]);
}

if(($store<0)&&($arr1[2]<0))
{
  $arr1[0]=0-$arr1[0];
  $arr1[2]=abs($arr1[2]);
}

if($arr1[0]%$arr1[2]==0)
{
  $quotient=$arr1[0]/$arr1[2];
  return ($quotient,"/",1);
}
}

for ($i=abs($arr1[0]*$arr1[2]); $i>1; $i--)
{
 if (($arr1[0]%$i==0)&&($arr1[2]%$i==0))
    { 
       $arr1[0]/=$i;
       $arr1[2]/=$i;
       $j=$i;
      
    }
}
return ($arr1[0],"/",$arr1[2]);
}

sub invert{

$temp=$arr1[0];
$arr1[0]=$arr1[2];
$arr1[2]=$temp;

return ($arr1[0], "/", $arr1[2]);
}

sub mult{

$temp1=$arr1[0]*$arr2[0];
$temp2=$arr1[2]*$arr2[2];

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

sub add{

if(($arr1[2]!=$arr2[2])&&(($arr1[2]!=0)&&($arr2[2]!=0)))
{
 for($count=1;$count<abs($arr1[2]*$arr2[2]);$count++)
  {if(($arr1[2]%$count==0)&&($arr2[2]%$count==0))
    {$gcf=$count;}
  }
$lcm=($arr1[2]*$arr2[2])/$gcf;
$temp=$lcm/$arr1[2];

$arr1[0]=($lcm/$arr1[2])*$arr1[0];
$arr2[0]=($lcm/$arr2[2])*$arr2[0];

$arr1[2]=($lcm/$arr1[2])*$arr1[2];
$arr2[2]=($lcm/$arr2[2])*$arr2[2];
}
$tempfrac1[0]=$arr1[0]+$arr2[0];
$tempfrac2[0]=$arr1[2];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub subtract{

$arr1[0]=($lcm/$arr1[2])*$arr1[0];
$arr2[0]=($lcm/$arr2[2])*$arr2[0];

$arr1[2]=($lcm/$arr1[2])*$arr1[2];
$arr2[2]=($lcm/$arr2[2])*$arr2[2];
}
$tempfrac1[0]=$arr1[0]-$arr2[0];
$tempfrac2[0]=$arr1[2];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub clean{

@array=();

@hold=@array;

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

$size=@array;

for($i=0;$i<3;$i++)
{
for($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]=~/\(/))&&($array[$count+2]=~/\-/))
 {delete $array[$count+2];
 $array[$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]=~/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];
}}

}}

if($form  eq 'parallel')
{
$string="@m";
$string=clean();
@cleanm=@array;

$string="y=(@m)*x+$b";
$string=clean();
@clean_start=@array;

print "

Since any two parallel lines have the same slope we know the slope of the unknown line is {{{@cleanm}}} (its from the slope of {{{@clean_start}}} which is also {{{@cleanm}}}).\nAlso since the unknown line goes through";

$string="@x1";
$string=clean();
@cleanx=@array;

$string="@y1";
$string=clean();
@cleany=@array;

print " (@cleanx,@cleany), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

{{{y-y[1]=m(x-x[1])}}} where m is the slope and ({{{x[1]}}},{{{y[1]}}}) is the given point

$string="y-@y1=(@m)*(x-@x1)";

$string=clean();

print "

{{{@array}}} Plug in {{{m=@cleanm}}}, {{{x[1]=@cleanx}}}, and  {{{y[1]=@cleany}}}

$string="y-@y1=(@m)*x-(@m)(@x1)";

$string=clean();

print "

{{{@array}}} Distribute {{{@cleanm}}}

";
@arr1=@m;
@arr2=@x1;
@multiply=mult();

$string="y-@y1=(@m)*x-@multiply";

$string=clean();

print "

{{{@array}}} Multiply

$string="y=(@m)*x-@multiply+@y1";

$string=clean();

print "

{{{@array}}}", ($y1[0]<0) ? "Subtract {{{@cleany}}} from both sides to isolate y" : "Add {{{@cleany}}} to both sides to isolate y";

if(($y1[2]==1)&&($multiply[2]==1))
{
 $add=$y1[0]-$multiply[0];
 $string="y=(@m)*x+$add";

$string=clean();
 @final=@array;
 $m2=$m[0]/$m[2];
 $b1=$add;

print "

{{{@array}}} Combine like terms

}

if(($y1[2]!=1)||($multiply[2]!=1))
{
@arr1=@y1;
@arr2=@multiply;
@sub=subtract();

@temp_storage1=@arr1;
@temp_storage2=@arr2;

$string="y=(@m)*x-@temp_storage2+@temp_storage1";

$string=clean();

print "

{{{@array}}} Make into equivalent fractions with equal denominators

$string="y=(@m)*x+@sub";

$string=clean();

print "

{{{@array}}} Combine the fractions

@arr1=@sub;
@sub=reduce();

$string="y=(@m)*x+@sub";

$string=clean();

@final=@array;
 $m2=$m[0]/$m[2];
 $b1=$sub[0]/$sub[2];

print "

{{{@array}}} Reduce any fractions

}

print " So the equation of the line that is parallel to {{{@clean_start}}} and goes through ({{{@cleanx}}},{{{@cleany}}}) is {{{@final}}}

So here are the graphs of the equations {{{@clean_start}}} and {{{@final}}}

{{{drawing( 500, 500, $xmin,$xmax,$ymin,$ymax, 
         grid(1),
         graph( 500, 500, $xmin,$xmax,$ymin,$ymax, $m1*x+$b, $m2*x+$b1),
         circle($xcoord,$ycoord,0.05),
         circle($xcoord,$ycoord,0.08)
)}}} graph of the given equation {{{@clean_start}}} (red) and graph of the line {{{@final}}}(green) that is parallel to the given graph and goes through ({{{@cleanx}}},{{{@cleany}}})

}

if($form  eq 'perpendicular')
{
@startm=@m;
$string="@m";
$string=clean();
@cleanm=@array;
@startm=@cleanm;

$string="y=(@m)*x+$b";
$string=clean();
@clean_start=@array;

print "

Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of {{{@startm}}}, you can find the perpendicular slope by this formula:

{{{m[p]=-1/m}}} where {{{m[p]}}} is the perpendicular slope

{{{m[p]=-1/(@m)}}} So plug in the given slope to find the perpendicular slope

@arr1=@m;
@m=invert();

print "

{{{m[p]=(-1/1)(@m)}}} When you divide fractions, you multiply the first fraction (which is really {{{1/1}}}) by the reciprocal of the second

$m[0]=-1*$m[0];
@arr1=@m;
@m=reduce();

$string="@m";
$string=clean();
@cleaninvertm=@array;

print "

{{{m[p]=@m}}} Multiply the fractions.

So the perpendicular slope is {{{@cleaninvertm}}}

So now we know the slope of the unknown line is {{{@cleaninvertm}}} (its the negative reciprocal of  {{{@cleanm}}} from the line {{{@clean_start}}}).\nAlso since the unknown line goes through";

$string="@x1";
$string=clean();
@cleanx=@array;

$string="@y1";
$string=clean();
@cleany=@array;

print " (@cleanx,@cleany), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

{{{y-y[1]=m(x-x[1])}}} where m is the slope and ({{{x[1]}}},{{{y[1]}}}) is the given point

$string="y-@y1=(@m)*(x-@x1)";

$string=clean();

print "

{{{@array}}} Plug in {{{m=@cleaninvertm}}}, {{{x[1]=@cleanx}}}, and  {{{y[1]=@cleany}}}

$string="y-@y1=(@m)*x-(@m)(@x1)";

$string=clean();

print "

{{{@array}}} Distribute {{{@cleaninvertm}}}

";
@arr1=@m;
@arr2=@x1;
@multiply=mult();

$string="y-@y1=(@m)*x-@multiply";

$string=clean();

print "

{{{@array}}} Multiply

$string="y=(@m)*x-@multiply+@y1";

$string=clean();

print "

{{{@array}}}", ($y1[0]<0) ? "Subtract {{{@cleany}}} from both sides to isolate y" : "Add {{{@cleany}}} to both sides to isolate y";

if(($y1[2]==1)&&($multiply[2]==1))
{
 $add=$y1[0]-$multiply[0];
 $string="y=(@m)*x+$add";

$string=clean();
 @final=@array;
 $m2=$m[0]/$m[2];
 $b1=$add;

print "

{{{@array}}} Combine like terms

}

if(($y1[2]!=1)||($multiply[2]!=1))
{
@arr1=@y1;
@arr2=@multiply;
@sub=subtract();

@temp_storage1=@arr1;
@temp_storage2=@arr2;

$string="y=(@m)*x-@temp_storage2+@temp_storage1";

$string=clean();

print "

{{{@array}}} Make into equivalent fractions with equal denominators

$string="y=(@m)*x+@sub";

$string=clean();

print "

{{{@array}}} Combine the fractions

@arr1=@sub;
@sub=reduce();

$string="y=(@m)*x+@sub";

$string=clean();

@final=@array;
 $m2=$m[0]/$m[2];
 $b1=$sub[0]/$sub[2];

print "

{{{@array}}} Reduce any fractions

}

print " So the equation of the line that is perpendicular to {{{@clean_start}}} and goes through ({{{@cleanx}}},{{{@cleany}}}) is {{{@final}}}

So here are the graphs of the equations {{{@clean_start}}} and {{{@final}}}

{{{drawing( 500, 500, $xmin,$xmax,$ymin,$ymax, 
         grid(1),
         graph( 500, 500, $xmin,$xmax,$ymin,$ymax, $m1*x+$b, $m2*x+$b1),
         circle($xcoord,$ycoord,0.05),
         circle($xcoord,$ycoord,0.08)
)}}} graph of the given equation {{{@clean_start}}} (red) and graph of the line {{{@final}}}(green) that is perpendicular to the given graph and goes through ({{{@cleanx}}},{{{@cleany}}})

}

==section output

==section check
  angle=40 angle1=50