Understanding Linear Algebra Systems | Maths Explanation for Perl Kids
Welcome to this beginner-friendly Perl tutorial designed for junior secondary students learning algebra. Perl is a great tool for solving math problems. It helps students visualize and automate algebraic solutions. This tutorial is perfect for:
- Junior secondary students learning Perl
- Teachers looking for math coding projects
- Anyone curious about solving equations with code
The resulting algorithm solves the linear system using basic algebra and Perl logic. It's ideal for school projects and math exercises.
Well, it hasn't been much of algebra in our junior secondary school Perl tutorial series so far; We are about to have our first taste of it.
What Are Simultaneous Equations? | Maths Explanation for Perl Kids
Simultaneous equations are a set of equations with multiple variables that are solved together. For example:
2x + 3y = 13; and
5x - y = 7
Simultaneous equations can be solved using algebraic methods like Substitution or Elimination,
or advanced methods like Matrix and Cramer's Rule.
In this tutorial, we'll focus on the elimination method and implement it using Perl.
Step-by-Step Guide to Solve Simultaneous Equations with 2 Unknowns | Elimination Method Perl Algorithm
Let's try to draft a Perl algorithm that solves simultaneous equations with 2 unknowns,
using the elimination method, with the following set of equations in consideration.
2x + 3y = 13; and
5x - y = 7
Step 1:
Pick a variable to eliminate, either x or y.
Our code will always eliminate the y variable.
Step 2:
Multiply equation 1 by the coefficient of variable y
in equation 2.
⇒
-1 X (2x + 3y = 13)
⇒
-2x - 3y = -13
Step 3:
Multiply equation 2 by the coefficient of variable y
in equation 1.
⇒
3 X (5x - y = 7)
⇒
15x - 3y = 21
Step 4:
Subtract the new equations obtained from Steps 2 and 3.
-2x - 3y = -13
-
15x - 3y = 21
⇒
-17x = -34
Step 5:
Divide the R.H.S. from Step 4 by the coefficient of
x to obtain x.
⇒
x = -34/-17 = 2;
Step 6:
Obtain y by solving for y from any of the
original equations, using the found value of x.
2x + 3y = 13
⇒
2(2) + 3y = 13;
⇒
3y = 13 - 4 = 9;
⇒
y = 9/3 = 3;
Create a new Perl module file;
call it Simultaneous2Unknown.pm.
Type out the adjoining Perl code for solving simultaneous equations with 2 unknowns.
Note: You can comment out the SortFraction Perl object
code in the main class from the previous lesson or simply continue from where it stopped.
So! Perl Fun Practice Exercise - Simultaneous Equations with 2 Unknowns
As a fun practice exercise, feel free to try out your own set of x_coefficients, y_coefficients and equals values, and see how the Perl code solves the resulting 2 by 2 Simultaneous Equations.
Perl Code for Solving Simultaneous Equations with 2 Unknowns - Module File
BEGIN {
require Exporter;
# for the sake of standard
our $VERSION = 2016.12;
# Inherit from exporter to export functions and variables
our @ISA = qw(Exporter);
# Functions and variables to be exported by default
our @EXPORT_OK = qw(solveSimultaneous);
}
use warnings;
use strict;
use Carp "croak";
my ($x_variable, $y_variable);
my (@x_coefficients, @y_coefficients, @equals);
my @eliminator;
my %equations;
# simulate an object construct
# takes one argument -- besides its package name;
# hash reference to array references for the variables
sub new {
no warnings "all";
my $this = shift;
my $parameters = shift; #this is already a hash reference
bless $parameters, $this;
$this->_init($parameters);
return $this;
}
# Simulate a constructor
sub _init {
my $self = shift;
my $aux = shift;
$equations{x} = $aux->{x};
$equations{y} = $aux->{y};
$equations{eq} = $aux->{eq};
@x_coefficients = @{$equations{x}};
@y_coefficients = @{$equations{y}};
@equals = @{$equations{eq}};
}
# Returns an array of the result
sub solveSimultaneous {
# this is actually an array of references
# not a multi-dimensional array
# STEP 2:
$eliminator[0][0] = $y_coefficients[1] * $x_coefficients[0];
$eliminator[0][1] = $y_coefficients[1] * $equals[0];
# STEP 3:
$eliminator[1][0] = $y_coefficients[0] * $x_coefficients[1];
$eliminator[1][1] = $y_coefficients[0] * $equals[1];
eval {
# STEPS 4, 5:
$x_variable = ($eliminator[0][1] - $eliminator[1][1]) / ($eliminator[0][0] - $eliminator[1][0]);
# STEP 6:
$y_variable = ($equals[0] - $x_coefficients[0] * $x_variable) / $y_coefficients[0];
return [$x_variable, $y_variable];
} or croak "Error $@ happened";
}
1;
Perl Code for Solving Simultaneous Equations with 2 Unknowns - Main Class
use strict;
use warnings;
use SIMULTANEOUS2UNKNOWN;
# Useful variables
my $solution;
my (@x_coefficients, @y_coefficients);
my (@operators, @equals);
##
# Simultaneous Equations with 2 unknowns
##
@operators = ('+', '+');
@x_coefficients = (2, 1);
@y_coefficients = (-1, 1);
@equals = (1, 2);
$operators[0] = '-' if $y_coefficients[0] < 0;
$operators[1] = '-' if $y_coefficients[1] < 0;
print "\n Solving simultaneously the equations:\n";
#Print as an equation
printf("%40dx %s %dy = %d\n", $x_coefficients[0], $operators[0], abs($y_coefficients[0]), $equals[0]);
printf("%40dx %s %dy = %d\n", $x_coefficients[1], $operators[1], abs($y_coefficients[1]), $equals[1]);
printf("\n%30s\n%40s", "Yields:", "(x, y) = ");
eval {
my $sim2unk = SIMULTANEOUS2UNKNOWN->new({
x => \@x_coefficients,
y => \@y_coefficients,
eq => \@equals
});
$solution = $sim2unk->solveSimultaneous();
printf("(%.4f, %.4f)\n", $solution->[0], $solution->[1]);
} or printf("(%s, %s)\n", "infinity", "infinity");
print "\n\n";