Jim is 42 years old. He has one brother, and their total age is 100. What is the brother’s age? OK, this isn’t a very hard problem, but let’s just introduce calc algebra by solving it.
M-x calc
'42 + x = 100 (' to enter algebraic input)
a S x (solve for x)
Result:
1: x = 58
Let’s make this harder. Jim and Dan’s ages sum to 100. Jim is 5 years older than Dan. How old are they?
'[j + d = 100, d + 5 = j] a S j,d Result: 1: [j = 52.5, d = 47.5]
Nice!
And of course it can give you more than just numerical solutions:
'sin(x) + tan(y) = pi / 2 a S y (solve for y) Result: 1: y = arctan(pi / 2 - sin(x))
Sometimes there are more than one solution. For example:
'x^2 = 25 a S x Result: 1: x = 5
Wait, what happened to -5! That’s a valid solution, why didn’t calc tell us about it? What’s happening here is that calc is telling us about the first valid thing it can find, which is basically how it operates. But you can always get everything:
'x^2 = 25 a P x (find the polynomial solutions) Result: 1: [5, -5]
Sometimes there aren’t a finite number of results because you aren’t dealing with polynomials. You can just get a generalized solution:
'sin(x)^2 = 25 H a S x (solve for x, giving the generalized solution) Result: 1: x = arcsin(5 s1) (-1)^n1 + 180 n1
This uses the calc notation n1, which you just means any integer. You can also
see another notation s1 which means any sign. In this case 5 s1 means that that
number can be 5 or -5.
Looking at how awesome calc is, it’s just a shame I never knew about it in high school…