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We generally credit Diophantus of Alexandria as laying the groundwork for algebra in the second or third centuries AD. We can't be certain because records from that time are a challenge to find. Use the link to the related question to read a bit on the history of algebra. You'll find that link below.
3.75 - without using a calculator !
To swap two variables without using a third variable, use exclusive or manipulation... a ^= b; b ^= a; a ^= b;
For elementary algebra equations, merely simplifying both sides and seeing if they match might work. For advanced algebra and trig, try transforming one side to the other, or both to a different third thing. If you are successful with either thing, it is true. If you reached the end of your figuring and got something that isn't always true (e.g. x2 = x or y = 5), then it's false.
Faciunt is the present indicative third-person plural of the verb facere, "to do; to make". It means "[they] do, make, are doing, are making".