$x=yx$. Can this statement be true when we don't know that $y=1$?

Question

I am dealing with an equation which is saying that $yx=x$. On the other hand it is telling us that $\frac{x}{x}=1$ which connotes that $x=x$. Is it not absurd to say that $x=x=yx$ when we don't know that $y=1$ here.

Answer 1

Dividing both sides of the equation by $x$ is valid provided $x\neq 0$. Note how doing so left out solution $x=0$. Instead, bring all terms to one side of the equation, factor, and solve.

$$x = yx \iff x - yx = 0 \iff x(1-y) = 0 \implies x=0 \text{ or } y = 1$$

Answer 2

$x = yx\implies yx - x = 0\implies (y - 1)x = 0\implies x = 0$ or $y = 1$.

Answer 3

Write the equation as $x(y-1)=0$. At least one factor must be zero, so that the solution is $$y=1\lor x=0.$$

Answer 4

Suppose that $x=0$, then $y$ must not necessarily be $1$, since any number multiplied by $0$ is $0$.

Suppose, instead, that $x\neq 0$, than you can divide both sides of the equation to get: $$\frac{x}{x}=y$$ and therefore $y=1$

I guess you're asking whether stating that $xy=x$ (when $x\neq 0)$ is absurd if we don't know, beforehand, that $y=1$. Well, it is if $y$ is already defined in some way and from that you deduce $y\neq 1$. But it is not if you are trying to find the set of solutions of the equation $xy=x$, where $x$ and $y$ are just arbitrary real numbers.

Answer 5

Are you talking about real numbers or integers? There are rings $R$ such that there are elements $a, b \neq 0$ with $ab=0$ (Take $\mathbb{Z}_6$ and $a=2, b=3$). In that case you cannot conclude $y=1$ from $xy=x$ for a single value of $x$.

In the example $y*3 = (y+2)*3$ for any $y$