Equation that is defined as an identity

[RChristenk]

Homework Statement

$5x-6=5x-6$

Relevant Equations

Algebraic concepts

$5x-6=5x-6$ is defined as an identity because it is true for all values of $x$.

My question is I can further simplify and arrive at $0=0$, in which case no values of $x$ will work because the variable $x$ itself doesn't exist.

Isn't this a contradiction? Or did I violate some rule when I simplified it to $0=0$?

 

[Mark44]

Homework Statement: $5x-6=5x-6$
Relevant Equations: Algebraic concepts

$5x-6=5x-6$ is defined as an identity because it is true for all values of $x$.

My question is I can further simplify and arrive at $0=0$, in which case no values of $x$ will work because the variable $x$ itself doesn't exist.

Isn't this a contradiction? Or did I violate some rule when I simplified it to $0=0$?

No, not a contradiction. The equation $5x - 6 = 5x - 6$ is equivalent to $0 = 0$. Like the first identity, the second is true for any values of x. For example, if x = 5, the equation 0 = 0 is still a true statement.

 

[FactChecker]

no values of $x$ will work

No. It means that any value of $x$ will work. No matter what value you set for $x$, you always have 0 = 0.

 

[PeroK]

For which values of $x$ is $0=0$ not true?

 

[WWGD]

Homework Statement: $5x-6=5x-6$
Relevant Equations: Algebraic concepts

$5x-6=5x-6$ is defined as an identity because it is true for all values of $x$.

My question is I can further simplify and arrive at $0=0$, in which case no values of $x$ will work because the variable $x$ itself doesn't exist.

Isn't this a contradiction? Or did I violate some rule when I simplified it to $0=0$?

It's also true for $x=\frac {6}{5}$