Somewhat stupid question, but why do the lines $y_1=x$ and $y_2=x+1$ never meet in a point?
Is it just because of algebra? Whatever $y_1$ is equal to, $y_2$ will be $y_1+1$?
Thanks.
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5$\begingroup$ Because they are paralell. Alegraically, the two equations have no common solutions. $\endgroup$Mauro ALLEGRANZA– Mauro ALLEGRANZA2026-04-23 08:34:50 +00:00Commented Apr 23 at 8:34
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$\begingroup$ It's because they're parallel. It's not an algebraic property, it's a geometric one. $\endgroup$CyclotomicField– CyclotomicField2026-04-23 08:37:39 +00:00Commented Apr 23 at 8:37
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1 Answer
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Suppose they could meet at some point $P(x_p,y_p)$, in other words there exists a point $P$ which belongs to both lines.
That is: the same point $P$ belongs to each of the two lines.
This means a pair of its coordinates $(x_p, y_p)$ satisfies both equations: $$y_p = x_p \quad \text{and} \quad y_p = x_p+1.$$
But equality is transitive: if two values equal to a third one, then the two are equal to each other.
So we have: $$x_p = y_p = x_p+1,$$ which results in $$x_p = x_p+1.$$
This, however, is not satisfied by any real value $x_p,$ because it would require $$0=1$$
which is false.
As a result, there is no such $x_p$ at which a point $P$ might exist belonging to both lines, hence lines do not have a common point—so they never meet.
