Simultaneous Equations With 3 Unknowns in VB.Net

3 by 3 Simultaneous Equations Code in Visual Basic (VB.Net)

To solve 3 by 3 simultaneous equations, we will simply eliminate the z variable, then call out to our simultaneous2unknown code module.

Algorithm Steps to Follow for Simultaneous Equations with 3 unknowns in VB.Net

Consider the equations:
         x + 2y - z = 2; and
         3x - y + 2z = 4
         2x + 3y + 4z = 9

Step 1:

Multiply equations 1, 2 & 3 by the LCM of the coefficients of variable z, divided by the z coefficient of the respective equation.
         (⁴/_-1) X (x + 2y - z = 2)
    ⇒     -4x - 8y + 4z = -8
         (⁴/₂) X (3x - y + 2z = 4)
    ⇒     6x - 2y + 4z = 8
         (⁴/₄) X (2x + 3y + 4z = 9)
    ⇒     2x + 3y + 4z = 9

Step 2:

Subtract the new equations obtained in Step 2; eqn (2) from eqn (1) and eqn (3) from eqn (2).
         -4x - 8y + 4z = -8
    -     6x - 2y + 4z = 8
    ⇒     -10x - 6y = -16
         6x - 2y + 4z = 8
    -     2x + 3y + 4z = 9     ⇒     4x - 5y = -1

Step 3:

Call out to our simultaneous2unknown code module to solve for x and y.
⇒ (x, y) = (1, 1);

Step 4:

Obtain z by solving for z from any of the original equations, using the found values of x and y.
         x + 2y - z = 2
⇒         1 + 2(1) - z = 2;
⇒         -z = 2 - 3 = -1;
⇒         z = ^-1/_-1 = 1;

Create a new class file; Project, Add Class.
Call it Simultaneous3Unknown.vb.
Optionally, create a new module file; Project, Add Module.
Call it Simultaneous3UnknownModule.vb.
Type out the adjoining Visual Basic (VB.Net) codes for solving simultaneous equations with 3 unknowns.

Note: The code module for finding LCM has been explained in the Primary Category.

You can instead comment out the previous VB.Net code from the main module or simply continue from where it stopped.

So!

Feel free to try out your own set of x_coefficients, y_coefficients, z_coefficients and equals values for 3 by 3 Simultaneous Equations.

VB.Net code for Simultaneous3Unknown Class File

Public Class Simultaneous3Unknown

    Dim x_coefficients() As Double
    Dim y_coefficients() As Double
    Dim z_coefficients() As Double
    Dim equal() As Double
    Dim eliminator(2, 2) As Double
    Dim x_variable As Double
    Dim y_variable As Double
    Dim z_variable As Double

    Dim lcm As Double

    ' A constructor
    Public Sub _init_(equations As Dictionary(Of String, Double()))
        x_coefficients = equations("x")
        y_coefficients = equations("y")
        z_coefficients = equations("z")
        equal = equations("eq")
    End Sub

    ' Returns a list Of the result
    Public Function solveSimultaneous() As Double()
        Dim l_c_m As New LCM
        l_c_m._init_({CInt(z_coefficients(0)), CInt(z_coefficients(1)), CInt(z_coefficients(1))})
        lcm = l_c_m.getLCM()

        ' STEP 1:
        ' eliminate z variable
        eliminator(0, 0) = (lcm * x_coefficients(0)) / z_coefficients(0)
        eliminator(0, 1) = (lcm * y_coefficients(0)) / z_coefficients(0)
        eliminator(0, 2) = (lcm * equal(0)) / z_coefficients(0)

        eliminator(1, 0) = (lcm * x_coefficients(1)) / z_coefficients(1)
        eliminator(1, 1) = (lcm * y_coefficients(1)) / z_coefficients(1)
        eliminator(1, 2) = (lcm * equal(1)) / z_coefficients(1)

        eliminator(2, 0) = (lcm * x_coefficients(2)) / z_coefficients(2)
        eliminator(2, 1) = (lcm * y_coefficients(2)) / z_coefficients(2)
        eliminator(2, 2) = (lcm * equal(2)) / z_coefficients(2)

        ' STEP 2:
        Dim new_x = {eliminator(0, 0) - eliminator(1, 0), eliminator(1, 0) - eliminator(2, 0)}
        Dim new_y = {eliminator(0, 1) - eliminator(1, 1), eliminator(1, 1) - eliminator(2, 1)}
        Dim new_eq = {eliminator(0, 2) - eliminator(1, 2), eliminator(1, 2) - eliminator(2, 2)}

        Try
            ' STEP 3
            Dim s2u As New Simultaneous2Unknown
            Dim aux As New Dictionary(Of String, Double())
            aux.Add("x", new_x)
            aux.Add("y", new_y)
            aux.Add("eq", new_eq)
            s2u._init_(aux)
            Dim partial_solution = s2u.solveSimultaneous()

            x_variable = partial_solution(0)
            y_variable = partial_solution(1)
            ' STEP 4:
            z_variable = (equal(0) - x_coefficients(0) * x_variable - y_coefficients(0) * y_variable) / z_coefficients(0)

            Return {x_variable, y_variable, z_variable}

        Catch ex As DivideByZeroException
            Throw ex
            Return Nothing
        End Try
    End Function

End Class

Main Module

Module Algebra_Simultaneous3Unknown

    Sub Main()
        ''
        ' Simultaneous Equations with 3 unknowns
        ''

        Dim x_coefficients As Double() = {2.0, 4.0, 2.0}
        Dim y_coefficients As Double() = {1.0, -1.0, 3.0}
        Dim z_coefficients As Double() = {1.0, -2.0, -8.0}
        Dim equal As Double() = {4.0, 1.0, -3.0}

        Dim equations As New Dictionary(Of String, Double())
        equations.Add("x", x_coefficients)
        equations.Add("y", y_coefficients)
        equations.Add("z", z_coefficients)
        equations.Add("eq", equal)

        Dim operators(2, 1) As String
        For i = 0 To 2
            operators(i, 0) = "+"
            If y_coefficients(i) < 0 Then
                operators(i, 0) = "-"
            End If
            operators(i, 1) = "+"
            If z_coefficients(i) < 0 Then
                operators(i, 1) = "-"
            End If
        Next i

        Console.WriteLine(vbCrLf, "    Solving simultaneously the equations:", vbCrLf)
        'Print as an equation
        Console.WriteLine(String.Format("{0,20}x  {1}  {2}y  {3}  {4}z  =  {5}",
                x_coefficients(0), operators(0, 0), Math.Abs(y_coefficients(0)),
                operators(0, 1), Math.Abs(z_coefficients(0)), equal(0)
            )
        )
        Console.WriteLine(String.Format("{0,20}x  {1}  {2}y  {3}  {4}z  =  {5}",
                x_coefficients(1), operators(1, 0), Math.Abs(y_coefficients(1)),
                operators(1, 1), Math.Abs(z_coefficients(1)), equal(1)
            )
        )
        Console.WriteLine(String.Format("{0,20}x  {1}  {2}y  {3}  {4}z  =  {5}",
                x_coefficients(2), operators(2, 0), Math.Abs(y_coefficients(2)),
                operators(2, 1), Math.Abs(z_coefficients(2)), equal(2)
            )
        )
        Console.WriteLine(Environment.NewLine)
        Console.Write(String.Format("{0,15} {1}{2,20}", "Yields:", vbCrLf, "(x, y, z)  =  "))

        Try
            Dim sim3unk As New Simultaneous3Unknown
            sim3unk._init_(equations)
            Dim solution = sim3unk.solveSimultaneous()

            Console.Write(String.Format("({0:0.0000}, {1:0.0000}, {2:0.0000})", solution(0), solution(1), solution(2)))

        Catch ex As Exception
            Console.Write("(infinity,  infinity, infinity)")
        End Try

    End Sub

End Module

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