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Some of the content of this guide was modeled after a guide originally created by the Openstax and has been adapted for the GPRC Learning Commons in September 2020. This work is licensed under a Creative Commons BY 4.0 International License .

Transposition is a method in mathematics to solve equations. In some mathematics problems, one has to solve an equation for a variable. In order to do that, one has to isolate the unknown variable. This process involves some inverse operations to undo the operations in the equation and get the unknown variable on one side of the equation by itself. The following inverse operations are often used in transposition: 
  • - undoes +
  • + undoes -
  • × undoes ÷
  • ÷  undoes × 
  • square undoes 
  •   undoes square.
Note that, according to the transposition method, one has to apply same operations to both sides of the equation.


Example 1): Solve 2x-5=8+4x.
In order to solve the equation, we have to isolate x:
Step 1)  We do not want to have -5 on the left hand side of the equation. To undo -5, we need to add 5 to both sides of the equation:  
                                         2x-5+5=8+4x+5   ⇒  2x=13+4x       
Step 2)  Now, we want to have x terms on one side of the equation. So, we subtract 4x from both sides:
                                         2x-4x=13+4x-4x  ⇒ -2x=13
Step 3) Finally, we need x by itself without the coefficient of -2. Therefore, we divide both sides of the equation by -2: 
                                         -2x ÷ (-2)=13 ÷ (-2) ⇒  x=-13/2.                                                    
Example 2): Solve 1-6x=12+5x.
Step 1) We subtract 1 from both sides:


                                                1-6x-1=12+5x-1 ⇒ -6x=11+5x


Step 2) We need to have x terms together on one side of the equation. Therefore, we subtract -5x from both sides:


                                                -6x-5x=11+5x-5x ⇒ -11x=11


Step 3) We divide both sides by -11 to have x by itself on the left hand side:




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