What does it mean when a Linear equation has no solutions?
Take for example the linear equation $$-7x+3=-7x+2$$
What I understand by this is that there is no number that would make the Left hand side and the right hand side of the equation equal.But if so, shouldn't there NOT be an "=" sign between the two expressions, because they cannot be equivalent to each other if they don't have a solution?
Shouldn't it also be understood this way?:
If the equation is simplified, we'd get: $3=2$which is not true. Hence,
$$-7x+3 \neq -7x+2$$
I'm confused.
$\endgroup$ 52 Answers
$\begingroup$An equation, even such as $2=3$, is a mathematical expression that has a truth value. This one is $\text{false}$.
When there is an unknown, the expression can be conditionally true: $3x-9=0$ is $\text{true}$ if and only if $x=3$ is $\text{true}$.
$3x-9=3x$ is never $\text{true}$, and this is said to be an impossible equation.
$\endgroup$ $\begingroup$Once it has been agreed among the humans of interest that the string $-7x+3=-7x+2$ is an equation in $x$, it is no longer to be interpreted as the statement (a):
(a) The right-hand side results from the left-hand side by means of algebraic manipulations.
But rather as part of the problem (b):
(b) Find the values of $x$ such that the identity holds.
And the algebraic manipulations are among lines which, ideally, should be linked by the connective of ligical equivalence $\Leftrightarrow$ (ore, more visually, $\Updownarrow$). Said link means "the values of $x$ such that the equation above holds are exactly the values of $x$ such that the equation below holds".
Sometimes, lines may be linked by connectives such as $\Downarrow$, which means "the values of $x$ that solve the equation above also solve the equation below". This, albeit useful when one wants to prove that an equation has no solutions, typically needs a subsequent check of which solutions of the latter, less restrictive equation are actually solutions of the former.
Of course, lines containing equations are allowed to be connected by explanatory text.
It is certiainly true that, since the equation $-7x+3=-7x+2$ has no solutions, then the inequation $-7x+3\ne -7x+2$ is solved by all $x$ (and it is therefore a full-fledged inequality, just like an equation solved by all $x$ is often called identity or equality).
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