Test this code, please. It it simpler, avoids Select, Activate, Copy and Paste and is very fast...
Private Sub CommandButton1_Click()
 Dim arr1 As Variant, arr2 As Variant, OutRangex2 As Range
 arr1 = Sheets("Line 1").Range("L4:L204").Value
 Sheets("Numeric Plot").Range("B1").Resize(1, UBound(arr1, 1)).Value = WorksheetFunction.Transpose(arr1)
 arr2 = Sheets("Line 3").Range("L4:L204").Value
 Set OutRangex2 = Sheets("Numeric Plot").Range("B1").End(xlToRight).Offset(0, 1)
  OutRangex2.Resize(1, UBound(arr2, 1)).Value = WorksheetFunction.Transpose(arr2)
End Sub
I think the duplicates removal should be consider like being object of a new question, according to our rules...
Please test the next code. It works also using arrays and should be very speedy. Please let me know how it works. It could be integrated in the first sub, but I made it from scratch...
Sub removeDuplicate()
 Dim arrSort As Variant, lastCol As Long, lastRow As Long, arrSorted As Variant, sh As Worksheet
  Set sh = Sheets("Numeric Plot")
  lastCol = sh.Cells(1, sh.Cells.Columns.count).End(xlToLeft).column 'last col on the first row
  arrSort = sh.Range(sh.Cells(1, 2), sh.Cells(1, lastCol)).Value        'put the row values in an array
  'transpose the array in a column after the last one of the rows 1:
  sh.Cells(1, lastCol + 1).Resize(UBound(arrSort, 2), 1).Value = WorksheetFunction.Transpose(arrSort)
  'remove duplicates with Excel function:
  sh.Range(sh.Cells(1, lastCol + 1), sh.Cells(UBound(arrSort, 2), lastCol + 1)).RemoveDuplicates Columns:=1, Header:=xlNo
  lastRow = sh.Cells(sh.Cells.Rows.count, lastCol + 1).End(xlUp).row 'Last row after dupplicate elimination
  arrSorted = sh.Range(sh.Cells(1, lastCol + 1), sh.Cells(lastRow, lastCol + 1)).Value 'The cleared column pun in an array
    sh.Range(sh.Cells(1, 2), sh.Cells(1, lastCol)).Clear                     'clearing the data of the first row
    sh.Range(sh.Cells(1, lastCol + 1), sh.Cells(lastRow, lastCol + 1)).Clear 'clearing the data of temporary column
    Dim finalRng As Range
     Set finalRng = sh.Range("B1").Resize(1, UBound(arrSorted))
    finalRng.Value = WorksheetFunction.Transpose(arrSorted) 'transpose the fiterred array
   'sort the resulted range:
   finalRng.Sort Key1:=finalRng, Order1:=xlAscending, Orientation:=xlLeftToRight
End Sub