The substitution property of equality is a part of its definition; you can substitute anywhere.
The substitution property of equality is a part of its definition; you can substitute anywhere.
Similarly, 1/3 = 0.3333…
So 3 times 1/3 = 0.9999… but also 3/3 = 1
Another nice one:
Let x = 0.9999… (multiply both sides by 10)
10x = 9.99999… (substitute 0.9999… = x)
10x = 9 + x (subtract x from both sides)
9x = 9 (divide both sides by 9)
x = 1
They did cooperate with authorities, but they also took their time in disclosures to explain precisely what the user did wrong, and how you can avoid making the same mistakes. At the end of the day, Proton only has the information you provide them. And if you don’t encrypt your stuff, it’s not safe.
No option to disable… that I found, that is.
Mine provides a connection, but doesn’t expose ports on v6. So I can access v6 services but can’t self-host any.
School teachers. Not deliberately, but there is a shortage of them here.
What do you mean “fuck them”, they don’t exist any more…
There’s a known Plasma 6 bug causing some weirdness at the bottom of the screen, but I expect it will be fixed soon. In any case, switching between virtual desktops gets rid of it.
Does M3/M4 support AV1 encoding?
You can use negative decibels, right? It’s a logarithmic scale
Before timezones and trains, each town had its own natural time (based on the sun or whatever). Would you have preferred that?
For any
a
,b
,c
, ifa = b
andb = c
, thena = c
, right? The transitive property of equality.For any
a
,b
,x
, ifa = b
, thenx + a = x + b
. The substitution property.By combining both of these properties, for any
a
,b
,x
,y
, ifa = b
andy = b + x
, it follows thatb + x = a + x
andy = a + x
.In our example,
a
isx'
(notice the'
) andb
is0.999…
(by definition).y
is10x'
andx
is9
. Let’s fill in the values.If
x' = 0.9999…
(true by definition) and10x = 0.999… + 9
(true by algebraic manipulation), then0.999… + 9 = x' + 9
and10x' = x' + 9
.If you actually change any of the sides. Since, after substitution, the numeric value doesn’t change (literally the definition of equality), I don’t have to do anything – as I’m not rearranging. I’m merely presenting the same value in an equivalent manner. By contrast, when multiplying both sides by 10, since multiplication by 10 changes the concrete numeric value, I have to do it on both sides to maintain the equality relation (ditto for subtracting
x'
). But substitution never changes a numeric value – only rearranges what we already know.(Edit)
Take the following simple system of equations.
How would you solve it? Here’s how I would:
Here’s how Microsoft Math Solver would do it.