Solver Graphing Linear Equations

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Algebra: Linear Equations, Graphs, Slope

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Source code of 'Graphing Linear Equations'

This Solver (Graphing Linear Equations) was created by by jim_thompson5910(13794)

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==section input
Please choose which form you will use *[choice form standard slope-intercept]. For instance if you have a problem similar to {{{2x+3y=5}}} then choose "standard". If you have something similar to {{{y=2x+5}}} then choose "slope-intercept"

Input the equation in standard form {{{Ax+By=C}}} *[input a=1]x+ *[input b=2]y= *[input c=3]

Enter your equation in slope-intercept form ({{{y=mx+b}}} form)

y=*[input m_input=2]x + *[input b_input=1]

note: make sure you choose the correct form from the drop box above. Also, if you have decimal coefficients, convert them to fractions. For instance, if you have {{{.2x+.1y=10}}}, then input it as {{{(1/5)x+(1/10)y=10}}}

==section solution

perl

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];
}}

}

#$string=@array;

$string=@hold;

}

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 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]);
}

if($form eq 'standard')
{

@a, @b, @c;

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

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

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

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

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

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

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

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

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

if($b[2]!=0)
{$b2=$b[0]/$b[2];}

if($a[2]!=0)
{$a2=$a[0]/$a[2];}

if($c[2]!=0)
{$c2=$c[0]/$c[2];}

if($b==0)
{@arr1=($c,"/",$a);
@fraction=reduce();
$string="@fraction";
$string=clean();

print "Since we have the equation {{{$a*x+$b*y=$c}}} we really have {{{$a*x=$c}}}

{{{$a*x=$c}}} Start with the given equation

{{{x=$c/$a}}} Divide both sides by $a

{{{x=@array}}} Reduce

So this means we have a vertical line at {{{x=@array}}}

{{{ graph( 300, 200, -6, 5, -10, 10, 1000(x-$c/$a)) }}}";
return undef;
}

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

$string=clean();

@initial=@array;

print "

{{{@array}}}Start with the given equation

@abs=@a;

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

$string="(@abs)*x";

$string=clean();

@abs=@array;

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

$string=clean();

$size=@array;

print "

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

@arr1=@b;
@temp_b=invert();

@arr1=@temp_b;
@temp_b=reduce();

$string="y=(@temp_b)(@c-(@a)*x)";

$string=clean();

print "

{{{@array}}} Multiply both sides by ";

$string="@temp_b";

$string=clean();

@distribute=@array;

print "{{{@distribute}}}";

$string="y=(@temp_b)(@c)-(@temp_b)(@a)x)";

$string=clean();

print "

{{{@array}}} Distribute {{{@distribute}}}";

@arr1=@temp_b;
@arr2=@c;
@temp_c=mult();

@arr1=@temp_b;
@arr2=@a;
@temp_a=mult();

$string="y=@temp_c-(@temp_a)x";

$string=clean();

print "

{{{@array}}} Multiply";

@arr1=("0","/",$temp_a[2]);
@arr2=@temp_a;
@temp_a=subtract();

$string="y=(@temp_a)*x+@temp_c";

$string=clean();

print "

{{{@array}}} Rearrange the terms";

@arr1=@temp_a;
@temp_a=reduce();

@arr1=@temp_c;
@temp_c=reduce();

$string="y=(@temp_a)*x+@temp_c";

$string=clean();

print "

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

@final=@array;

print "

So the equation is now in slope-intercept form ({{{y=mx+b}}}) where ";

$string="@temp_a";

$string=clean();

$slope1="@temp_a";

if($slope1=~/\//)
{@slope=split(/\//,$slope1);}

if($slope1!~/\//)
{$slope .="/1";
@slope=split(/\//,$slope1);}

@slope=($slope[0],"/",$slope[1]);

print "{{{m=@array}}} (the slope) and ";

$string="@temp_c";

$string=clean();

@arr1=@temp_c;
@y_int=reduce();

print "{{{b=@array}}} (the y-intercept)";

print "

So to graph this equation lets plug in some points";

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

$int=$temp_c[0]/$temp_c[2];

$xint=(0-$int)/($m);

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

@arr1=("0","/",$temp_c[2]);
@arr2=@temp_c;
@x_int=subtract();

@arr1=@x_int;
@arr2=@m;
@x_int=mult();

@arr1=@x_int;
@x_int=reduce();

$tick=0;

if(($temp_a[0]%$temp_a[2]!=0)&&($temp_c[0]%$temp_c[2]!=0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

#abs($y_temp1[0]+$y_temp1[1])/2

if(($temp_a[0]%$temp_a[2]!=0)&&($temp_c[0]%$temp_c[2]==0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

if(($temp_a[0]%$temp_a[2]==0)&&($temp_c[0]%$temp_c[2]!=0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

$tick=0;

if(($temp_a[0]%$temp_a[2]==0)&&($temp_c[0]%$temp_c[2]==0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

$store=abs($x_max-$x_min);
$store1=($y_max-$y_min);
$rad=$store/150;

if(length($x[0])==0)
{$x[0]=0;}

if((length($x[1])==0)&&($x[0]!=1))
{$x[1]=1;}

if((length($x[1])==0)&&($x[0]==1))
{$x[1]=0;}

if($x[0]<$x[1])
{$x_min=$x[0]-10;
 $x_max=$x[1]+10;
}

if($x[1]<$x[0])
{$x_min=$x[1]-10;
 $x_max=$x[0]+10;
}

$y_temp1[0]=$m*$x[0]+$int;
$y_temp1[1]=$m*$x[1]+$int;

if($y_temp1[0]<$y_temp1[1])
{$y_min=$y_temp1[0]-10;
 $y_max=$y_temp1[1]+10;
}

if($y_temp1[1]<$y_temp1[0])
{$y_min=$y_temp1[1]-10;
 $y_max=$y_temp1[0]+10;
}

print "

Plug in x=$x[0]";

$string="y=(@temp_a)*($x[0])+@temp_c";

$string=clean();

print "

{{{@array}}}";

@mult_a=($x[0], "/",1);
@arr1=@temp_a;
@arr2=@mult_a;
@mult_a=mult();

$string="y=@mult_a+@temp_c";

$string=clean();

print "

{{{@array}}} Multiply";

@arr1=@mult_a;
@arr2=@temp_c;
@add_c=add();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Add";

if($add_c[2]!=1)
{
@arr1=@add_c;
@add_c=reduce();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Reduce";}

$size=@add_c;

if($size>1)
{$y[2]=$add_c[0]/$add_c[2];}

if($size==1)
{$y[2]=$add_c[0]/$add_c[2];}

print "

So here's one point ($x[0],$y[2])

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}}

print "Now lets find another point

Plug in x=$x[1]";

$string="y=(@temp_a)*($x[1])+@temp_c";

$string=clean();

print "

{{{@array}}}";

@mult_a=($x[1], "/",1);
@arr1=@temp_a;
@arr2=@mult_a;
@mult_a=mult();

$string="y=@mult_a+@temp_c";

$string=clean();

print "

{{{@array}}} Multiply";

@arr1=@mult_a;
@arr2=@temp_c;
@add_c=add();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Add";

if($add_c[2]!=1)
{
@arr1=@add_c;
@add_c=reduce();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Reduce";}

$size=@add_c;
if($size>1)
{$y[4]=$add_c[0]/$add_c[2];}

if($size==1)
{$y[4]=$add_c[0]/$add_c[2];}

print "

So here's another point ($x[1],$y[4]). Add this to our graph

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}}

Now draw a line through these points";

#$store=$m*(-10)+$int;
#$store=$m*(10)+$int;

print "

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}} So this is the graph of {{{@final}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])

So from the graph we can see that the slope is {{{@slope}}} (which tells us that in order to go from point to point we have to start at one point and ", ($slope[0]>0) ? "go  up $slope[0] units and to the right $slope[2] units to get to the next point)": "go  down $slope[0] units and to the right $slope[2] units to get to the next point),";

print " the y-intercept is (0,{{{$int}}})";

if($y_int[2]!=1)
{print " ,or (0,{{{@y_int}}}), ";}

print "and the x-intercept is ({{{$xint}}},0) ";

if($x_int[2]!=1)
{print " ,or ({{{@x_int}}},0) "};

print ". So all of this information verifies our graph.";

$string="@y_int";
$string=clean();
@b_array=@array;

$x[0]=0;
$y[2]=$m*(0)+$int;

$x[1]=$slope[2];
$y[4]=$slope[0]+$y[2];

$x_min=$x[0]-10;
$x_max=$x[1]+10;

if($y[2]>$y[4])
{$y_min=$y[4]-10;
$y_max=$y[2]+10;}

if($y[4]>$y[2])
{$y_min=$y[2]-10;
$y_max=$y[4]+10;}

$store=abs($x_max-$x_min);
$store1=($y_max-$y_min);
$rad=$store/150;

$temp_abs=abs($slope[0]);

print "

We could graph this equation another way. Since {{{b=@b_array}}} this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,{{{@b_array}}}).

So we have one point (0,{{{@b_array}}})

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),

blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}}

Now since the slope is {{{@slope}}}, this means that in order to go from point to point we can use the slope to do so. So starting at (0,{{{@b_array}}}), we can go ", ($slope[0]>0)? "up $temp_abs units ":"down $temp_abs units ";

print "

{{{drawing(  600, 600, $x_min,$x_max, $y_min, $y_max,
  grid(1),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) ),
blue( arc( 0, $y[2]+($slope[0]/2), 2, $slope[0], 90, 270 ) )

)}}}";

$x[1]=$slope[2];
$y[4]=$slope[0]+$y[2];

if($slope[0]>0)
{print "
and to the right $slope[2] units to get to our next point

{{{drawing(  600, 600, $x_min,$x_max, $y_min, $y_max,
  grid(1),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) ),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
blue( arc( 0, $y[2]+($slope[0]/2), 2, $slope[0], 90, 270 ) ),
blue( arc( ($slope[2]/2), $y[4], $slope[2], 2, 180, 360 ) )

)}}}";

print "

Now draw a line through those points to graph {{{@final}}}

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) ),
blue( arc( 0, $y[2]+($slope[0]/2), 2, $slope[0], 90, 270 ) ),
blue( arc( ($slope[2]/2), $y[4], $slope[2], 2, 180, 360 ) )
)}}} So this is the graph of {{{@final}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])
";
}

if($slope[0]<0)
{print "
and to the right $slope[2] units to get to our next point

)}}}";

print "

Now draw a line through those points to graph {{{@final}}}

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) ),
blue( arc( 0, $y[2]+($slope[0]/2), 2, $slope[0], 90, 270 ) ),
blue( arc( ($slope[2]/2), $y[4], $slope[2], 2, 0, 180 ) )
)}}} So this is the graph of {{{@final}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])
";
}

}

if($form eq 'slope-intercept')
{

@m_input, @b_input;

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

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

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

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

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

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

@temp_a=@m_input;
@temp_c=@b_input;

$string="y=(@m_input)*x+@b_input";

$string=clean();

@initial=@array;

print "In order to graph {{{@initial}}} we only need to plug in two points to draw the line

So lets plug in some points";

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

$int=$temp_c[0]/$temp_c[2];

$xint=(0-$int)/($m);

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

@arr1=("0","/",$temp_c[2]);
@arr2=@temp_c;
@x_int=subtract();

@arr1=@x_int;
@arr2=@m;
@x_int=mult();

@arr1=@x_int;
@x_int=reduce();

$tick=0;

$y_temp1[0]=$m*$x[0]+$int;
$y_temp1[1]=$m*$x[1]+$int;

if(($temp_a[0]%$temp_a[2]!=0)&&($temp_c[0]%$temp_c[2]!=0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

if(($temp_a[0]%$temp_a[2]!=0)&&($temp_c[0]%$temp_c[2]==0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

if(($temp_a[0]%$temp_a[2]==0)&&($temp_c[0]%$temp_c[2]!=0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

$y_temp1[0]=$m*$x[0]+$int;
$y_temp1[1]=$m*$x[1]+$int;

$tick=0;

if(($temp_a[0]%$temp_a[2]==0)&&($temp_c[0]%$temp_c[2]==0))
{for($i=-9;$i<=9;$i++)
{
 $temp=$m*$i+$int;
 if(($temp!~/\./)&&(abs($temp)<10))
  {$x[$tick]=$i;
    $tick++;}
}
}

if(length($x[0])==0)
{$x[0]=0;}

if((length($x[1])==0)&&($x[0]!=1))
{$x[1]=1;}

if((length($x[1])==0)&&($x[0]==1))
{$x[1]=0;}

if($x[0]<$x[1])
{$x_min=$x[0]-10;
 $x_max=$x[1]+10;
}

if($x[1]<$x[0])
{$x_min=$x[1]-10;
 $x_max=$x[0]+10;
}

$y_temp1[0]=$m*$x[0]+$int;
$y_temp1[1]=$m*$x[1]+$int;

if($y_temp1[0]<$y_temp1[1])
{$y_min=$y_temp1[0]-10;
 $y_max=$y_temp1[1]+10;
}

if($y_temp1[1]<$y_temp1[0])
{$y_min=$y_temp1[1]-10;
 $y_max=$y_temp1[0]+10;
}

$store=abs($x_max-$x_min);
$store1=($y_max-$y_min);
$rad=$store/150;

print "

Plug in x=$x[0]";

$string="y=(@temp_a)*($x[0])+@temp_c";

$string=clean();

print "

{{{@array}}}";

@mult_a=($x[0], "/",1);
@arr1=@temp_a;
@arr2=@mult_a;
@mult_a=mult();

$string="y=@mult_a+@temp_c";

$string=clean();

print "

{{{@array}}} Multiply";

@arr1=@mult_a;
@arr2=@temp_c;
@add_c=add();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Add";

if($add_c[2]!=1)
{
@arr1=@add_c;
@add_c=reduce();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Reduce";}

$size=@add_c;

if($size>1)
{$y[2]=$add_c[0]/$add_c[2];}

if($size==1)
{$y[2]=$add_c[0]/$add_c[2];}

print "

So here's one point ($x[0],$y[2])

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}}

print "Now lets find another point

Plug in x=$x[1]";

$string="y=(@temp_a)*($x[1])+@temp_c";

$string=clean();

print "

{{{@array}}}";

@mult_a=($x[1], "/",1);
@arr1=@temp_a;
@arr2=@mult_a;
@mult_a=mult();

$string="y=@mult_a+@temp_c";

$string=clean();

print "

{{{@array}}} Multiply";

@arr1=@mult_a;
@arr2=@temp_c;
@add_c=add();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Add";

if($add_c[2]!=1)
{
@arr1=@add_c;
@add_c=reduce();

$string="y=@add_c";

$string=clean();

print "

{{{@array}}} Reduce";}

$size=@add_c;
if($size>1)
{$y[4]=$add_c[0]/$add_c[2];}

if($size==1)
{$y[4]=$add_c[0]/$add_c[2];}

print "

So here's another point ($x[1],$y[4]). Add this to our graph

Now draw a line through these points";

#$store=$m*(-10)+$int;
#$store=$m*(10)+$int;

print "

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], .15, 1.5 )),
  blue( circle( $x[1],$y[4], .1, 1.5 ) ),
  blue( circle( $x[0],$y[2], .15, 1.5 )),
  blue( circle( $x[0],$y[2], .1, 1.5 ) )
)}}} So this is the graph of {{{@initial}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])

So from the graph we can see that the slope is {{{@m_input}}} (which tells us that in order to go from point to point we have to start at one point and ", ($m_input[0]>0) ? "go  up $m_input[0] units and to the right $m_input[2] units to get to the next point)": "go  down $m_input[0] units and to the right $m_input[2] units to get to the next point),";

print " the y-intercept is (0,{{{$int}}})";

if($b_input[2]!=1)
{print " ,or (0,{{{@b_input}}}), ";}

print "and the x-intercept is ({{{$xint}}},0) ";

if($x_int[2]!=1)
{print " ,or ({{{@x_int}}},0) "};

$string="@b_input";
$string=clean();
@b_array=@array;

$string="@m_input";
$string=clean();
@m_array=@array;

$y[2]=$b_input[0]/$b_input[2];
$x[0]=0;

$x[1]=$m_input[2];
$y[4]=$m_input[0]+$y[2];

$x_min=$x[0]-10;
$x_max=$x[1]+10;

if($y[2]>$y[4])
{$y_min=$y[4]-10;
$y_max=$y[2]+10;}

if($y[4]>$y[2])
{$y_min=$y[2]-10;
$y_max=$y[4]+10;}

$store=abs($x_max-$x_min);
$store1=($y_max-$y_min);
$rad=$store/150;
$temp_abs=abs($m_input[0]);

print "

We could graph this equation another way. Since {{{b=@b_array}}} this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,{{{@b_array}}}).

So we have one point (0,{{{@b_array}}})

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),

blue( circle( $x[0],$y[2], $rad)),
  blue( circle( $x[0],$y[2], $rad-0.05) )
)}}}

Now since the slope is {{{@m_input}}}, this means that in order to go from point to point we can use the slope to do so. So starting at (0,{{{@b_array}}}), we can go ", ($m_input[0]>0)? "up $temp_abs units ":"down $temp_abs units ";

print "

{{{drawing(  600, 600, $x_min,$x_max, $y_min, $y_max,
  grid(1),
  blue( circle( $x[0],$y[2], $rad)),
  blue( circle( $x[0],$y[2], $rad-0.05) )
blue( arc( 0, $y[2]+($m_input[0]/2), 2, $m_input[0], 90, 270 ) )

)}}}";

if($m_input[0]>0)
{print "
and to the right $m_input[2] units to get to our next point

{{{drawing(  600, 600, $x_min,$x_max, $y_min, $y_max,
  grid(1),
  blue( circle( $x[1],$y[4], $rad)),
  blue( circle( $x[1],$y[4], $rad-0.05 ) ),
  blue( circle( $x[0],$y[2], $rad)),
  blue( circle( $x[0],$y[2], $rad-0.05) ),
blue( arc( 0, $y[2]+($m_input[0]/2), 2, $m_input[0], 90, 270 ) ),
blue( arc( ($m_input[2]/2), $y[4], $m_input[2], 2, 180, 360 ) )

)}}}";

print "
Now draw a line through those points to graph {{{@initial}}}

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], $rad)),
  blue( circle( $x[1],$y[4], $rad-0.05 ) ),
  blue( circle( $x[0],$y[2], $rad)),
  blue( circle( $x[0],$y[2], $rad-0.05) ),
blue( arc( 0, $y[2]+($m_input[0]/2), 2, $m_input[0], 90, 270 ) ),
blue( arc( ($m_input[2]/2), $y[4], $m_input[2], 2, 180, 360 ) )
)}}} So this is the graph of {{{@initial}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])
";
}

if($m_input[0]<0)
{print "
and to the right $m_input[2] units to get to our next point

)}}}";

print "
Now draw a line through those points to graph {{{@initial}}}

{{{drawing( 600, 600, $x_min,$x_max, $y_min, $y_max,
  grid( 1 ),
  line($x_min-10,$m*($x_min-10)+$int,$x_max+10,$m*($x_max+10)+$int),
  blue( circle( $x[1],$y[4], $rad)),
  blue( circle( $x[1],$y[4], $rad-0.05 ) ),
  blue( circle( $x[0],$y[2], $rad)),
  blue( circle( $x[0],$y[2], $rad-0.05) ),
blue( arc( 0, $y[2]+($m_input[0]/2), 2, $m_input[0], 90, 270 ) ),
blue( arc( ($m_input[2]/2), $y[4], $m_input[2], 2, 0, 180 ) )
)}}} So this is the graph of {{{@initial}}} through the points ($x[0],$y[2]) and ($x[1],$y[4])
";
}

}

==section output

==section check
  angle=40 angle1=50