Why do we either add or subtract the same number infront of the equal sign?



With rule one applied:


But why should you do this in order to solve a relatively easy equation like $x+9=14$?


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  • $\begingroup$ How else would you solve the equation? $\endgroup$ – 5xum Aug 17 '17 at 10:28
  • $\begingroup$ @5xum by $14-9=x$? $\endgroup$ – yokihadu Aug 17 '17 at 10:28
  • $\begingroup$ And how do you know that $14-9=x$? $\endgroup$ – 5xum Aug 17 '17 at 10:29
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    $\begingroup$ Why don't you substract $9$ instead of add $3$? $\endgroup$ – Robert Z Aug 17 '17 at 10:29
  • 1
    $\begingroup$ @yokihadu Can you explain exactly what you would do? Make a step by step explanation, and justify each step. $\endgroup$ – 5xum Aug 17 '17 at 10:33

You would say that it is very obvious that the solution is


but why do you think you are allowed to just move the $9$ to the other side with a minus sign? Well, you are correct, you can do this. But the justification needs your first rule. Actually you do the following:

\begin{align} x+9&=14&&|\;-\!9\text{ on both sides}\\ x+\color{red}{9-9}&=14-9 &&|\;\color{red}{9-9}=0 \\ x + 0&= 14-9 \\ x &=5. \end{align}

But as you learned that this is a very long way to do a very easy thing, you just drop the intermediate steps and directly write the $9$ to the other side.


It´s a way to isolate the x that in the case of equations of degree 1 works very mechanically. Basically, to do it formally, I guess that you shall find a solution and prove that it´s unique. Well it´s proved that an equation of degree 1 has an unique solution on $\mathbb{C}$ so you just have to find the solution in the way you want. You can do it even trying.


The question of yours that why we are adding or subtracting same number on the both sides of an equation is answered by considering the fact that we have to find the value of x which satisfies the equation. Therefore, we employ a trick to isolate x on one side of equation.

The question that how this act of adding or subtracting is justified needs a proof which you can easily construct yourself. If you face any problems, post them here.

The only thing which I will advice you is to use the truth of axioms to construct your proof. These axioms may be of real number system if you take them as undefined object or if you take, for example positive integers as undefined objects, then use validity of theorems related to real numbers deduced from axioms to construct a proof.


You would do so that $x$ is the only term on the left hand side.

Solving an equation means getting the term you want to solve for (e.g $x$) on its own on one side of the equation.

If you have an equation of the form $x+a=b$, then to get $x$ on its own, you need to subtract $a$ from both sides. Subtracting any other number that is not equal to $a$, will not make $x$ the only term on the left hand side


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