Enhanced Simultaneous Mixed Equations Solver

Simultaneous equations involve two or more unknown variables. To solve them, you need as many equations as the number of unknown variables.

This calculator solves mixed simultaneous equations - one linear equation and one quadratic equation. The equations take the following form:

ax + by = m

cx² + dy² = n

The solution is found using the substitution method. We express one variable in terms of the other from the linear equation, then substitute into the quadratic equation to solve.

Interactive Equation Solver

Enter the coefficients for your equations and click "Solve" to get the solutions:

a:

b:

m:

c:

d:

n:

Solution 1

x₁ = ?

y₁ = ?

Solution 2

x₂ = ?

y₂ = ?

Graphical Representation

The solutions represent the points where the line intersects the quadratic curve

(x₁, y₁)

(x₂, y₂)

How It Works

To solve the mixed simultaneous equations:

ax + by = m

cx² + dy² = n

We follow these steps:

Express y in terms of x from the linear equation: y = (m - ax)/b
Substitute this expression into the quadratic equation
Solve the resulting quadratic equation in x
For each solution of x, compute the corresponding y value

The formulas used in the solution are:

x₁, x₂ = [m·a·d ± √(m²·a²·d² - (b²·c + a²·d)(d·m² - b²·n)] / (b²·c + a²·d)

y = (m - a·x)/b

Implementation Code

The original VB6 code for solving mixed simultaneous equations:

Private Sub Command1_Click()
    Dim a, b, c, d, m, n As Integer
    Dim x1, x2, y1, y2 As Double
    
    a = Val(Txt_a.Text)
    b = Val(Txt_b.Text)
    m = Val(Txt_m.Text)
    c = Val(Txt_c.Text)
    d = Val(Txt_d.Text)
    n = Val(Txt_n.Text)
    
    ' Calculate x1 and x2
    x1 = (m * a * d + Sqr(m ^ 2 * a ^ 2 * d ^ 2 - _
        (b ^ 2 * c + a ^ 2 * d) * (d * m ^ 2 - b ^ 2 * n))) / _
        (b ^ 2 * c + a ^ 2 * d)
        
    x2 = (m * a * d - Sqr(m ^ 2 * a ^ 2 * d ^ 2 - _
        (b ^ 2 * c + a ^ 2 * d) * (d * m ^ 2 - b ^ 2 * n))) / _
        (b ^ 2 * c + a ^ 2 * d)
    
    ' Calculate corresponding y values
    y1 = (m - a * x1) / b
    y2 = (m - a * x2) / b
    
    ' Display results rounded to 2 decimal places
    Lbl_x1.Caption = Round(x1, 2)
    Lbl_y1.Caption = Round(y1, 2)
    Lbl_x2.Caption = Round(x2, 2)
    Lbl_y2.Caption = Round(y2, 2)
End Sub

Equivalent VB.NET code with improved error handling:

Private Sub SolveButton_Click(ByVal sender As Object, _
                            ByVal e As EventArgs) Handles SolveButton.Click
    
    ' Parse input values with validation
    Dim a As Double = ParseDouble(txtA.Text)
    Dim b As Double = ParseDouble(txtB.Text)
    Dim m As Double = ParseDouble(txtM.Text)
    Dim c As Double = ParseDouble(txtC.Text)
    Dim d As Double = ParseDouble(txtD.Text)
    Dim n As Double = ParseDouble(txtN.Text)
    
    If b = 0 Or (b*b*c + a*a*d) = 0 Then
        MessageBox.Show("Invalid coefficients - division by zero")
        Return
    End If
    
    ' Calculate discriminant
    Dim discriminant As Double = m*m*a*a*d*d - _
                       (b*b*c + a*a*d) * (d*m*m - b*b*n)
    
    If discriminant < 0 Then
        MessageBox.Show("No real solutions exist")
        Return
    End If
    
    ' Calculate solutions
    Dim denominator As Double = b*b*c + a*a*d
    Dim x1 As Double = (m*a*d + Math.Sqrt(discriminant)) / denominator
    Dim x2 As Double = (m*a*d - Math.Sqrt(discriminant)) / denominator
    
    Dim y1 As Double = (m - a*x1) / b
    Dim y2 As Double = (m - a*x2) / b
    
    ' Display results
    lblX1.Text = Math.Round(x1, 2).ToString()
    lblY1.Text = Math.Round(y1, 2).ToString()
    lblX2.Text = Math.Round(x2, 2).ToString()
    lblY2.Text = Math.Round(y2, 2).ToString()
End Sub

Private Function ParseDouble(ByVal input As String) As Double
    Dim result As Double
    If Not Double.TryParse(input, result) Then
        MessageBox.Show("Invalid number format")
        Return 0
    End If
    Return result
End Function

The VB.NET version includes improved error handling, validation, and uses modern .NET math functions.

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