A1*x1 + B1*x2 + ... + N1*xn = c1
    A2*x1 + B2*x2 + ... + N2*xn = c2
    ...
    Am*x1 + Bm*x2 + ... + Nm*xn = cm = 0

For example:

    9 * x1 + 4 * x2 = 7
    4 * x1 + 3 * x2 = 8

You can write these equations as a matrix multiplied by a vector of variables x1, x2, ..., xn, equals a vector of constants c1, c2, ..., cn.

    | A1  B1  ...  N1 |   | x1 |   | c1 |
    | A2  B2  ...  N2 |   | x2 |   | c2 |
    |         ...     | * |... | = |... |
    | Am  Bm  ...  Nm |   | xm |   | cm |

Subroutine LoadArray loads data from text boxes into an augmented matrix that includes the constants:

    | A1  B1  ...  N1  c1 |
    | A2  B2  ...  N2  c2 |
    |         ...         |
    | Am  Bm  ...  Nm  cm |

' Load the augmented array.
' Column num_cols + 1 holds the result values.
' Column num_cols + 2 will hold the variables' final values.
Private Function LoadArray(ByRef num_rows As Integer, ByRef _
    num_cols As Integer) As Double(,)
    Dim value_rows() As String
    Dim coef_rows() As String
    Dim one_row() As String
    Dim arr(,) As Double

    ' Build the augmented matrix.
    value_rows = Split(txtValues.Text, vbCrLf)
    coef_rows = Split(txtCoefficients.Text, vbCrLf)
    one_row = Split(coef_rows(0), " ")
    num_rows = coef_rows.GetUpperBound(0) + 1
    num_cols = one_row.GetUpperBound(0) + 1
    ReDim arr(num_rows - 1, num_cols + 1)
    For r As Integer = 0 To num_rows - 1
        one_row = Split(coef_rows(r), " ")
        For c As Integer = 0 To num_cols - 1
            arr(r, c) = CDbl(one_row(c))
        Next c
        arr(r, num_cols) = CDbl(value_rows(r))
    Next r

    Return arr
End Function

Next the program uses row 2 to zero out the second column in rows 3 and later. It continues in this way until it can no longer zero out rows. Ideally at that point, the final row has the form | 0 0 ... Km Lm |. This represents the equation 0*x1 + 0*x2 + ... + Km*xm = Lm so it is easy to solve for xm. The program plugs that value into the second-to-last row to calculate x(m-1) and continues "backsolving" up the list until it has a value for every variable.

Private Sub cmdSolve_Click(ByVal eventSender As _
    System.Object, ByVal eventArgs As System.EventArgs) _
    Handles cmdSolve.Click
    Const TINY As Double = 0.00001
    Dim num_rows As Integer
    Dim num_cols As Integer
    Dim tmp As Double
    Dim factor As Double
    Dim arr(,) As Double
    Dim orig_arr(,) As Double
    Dim txt As String

    ' Build the augmented matrix.
    arr = LoadArray(num_rows, num_cols)
    orig_arr = LoadArray(num_rows, num_cols)
    
    ' Display the initial array.
    'PrintArray(arr)
    'PrintArray(orig_arr)

    ' Start solving.
    For r As Integer = 0 To num_rows - 2
        ' Zero out all entries in column r after this row.
        ' See if this row has a non-zero entry in column r.
        If Abs(arr(r, r)) < TINY Then
            ' Not a non-zero value. Try to swap with a
            ' later row.
            For r2 As Integer = r + 1 To num_rows - 1
                If Abs(arr(r2, r)) > TINY Then
                    ' This row will work. Swap them.
                    For c As Integer = 0 To num_cols
                        tmp = arr(r, c)
                        arr(r, c) = arr(r2, c)
                        arr(r2, c) = tmp
                    Next c
                    Exit For
                End If
            Next r2
        End If

        ' If this row has a non-zero entry in column r,
        ' skip this column.
        If Abs(arr(r, r)) > TINY Then
            ' Zero out this column in later rows.
            For r2 As Integer = r + 1 To num_rows - 1
                factor = -arr(r2, r) / arr(r, r)
                For c As Integer = r To num_cols
                    arr(r2, c) = arr(r2, c) + factor * _
                        arr(r, c)
                Next c
            Next r2
        End If
    Next r

    ' Display the upper-triangular array.
    'PrintArray(arr)

    ' See if we have a solution.
    If arr(num_rows - 1, num_cols - 1) = 0 Then
        ' We have no solution.
        txt = "No solution"
    Else
        ' Back solve.
        For r As Integer = num_rows - 1 To 0 Step -1
            tmp = arr(r, num_cols)
            For r2 As Integer = r + 1 To num_rows - 1
                tmp -= arr(r, r2) * arr(r2, num_cols + 1)
            Next r2
            arr(r, num_cols + 1) = tmp / arr(r, r)
        Next r

        ' Display the results.
        txt = "    Values"
        For r As Integer = 0 To num_rows - 1
            txt &= vbCrLf & "x" & r.ToString & " = " & _
                arr(r, num_cols + 1).ToString
        Next r

        ' Verify.
        txt &= vbCrLf & "    Check:"
        For r As Integer = 0 To num_rows - 1
            tmp = 0
            For c As Integer = 0 To num_cols - 1
                tmp += orig_arr(r, c) * arr(c, num_cols + 1)
            Next c
            txt &= vbCrLf & tmp.ToString
        Next r

        txt = txt.Substring(vbCrLf.Length + 1)
    End If

    txtResults.Text = txt
End Sub