This section provides algebraic calculation example of adding two distinct points on an elliptic curve.

Now we have algebraic formulas to calculate the addition operation on elliptic curves. Let's try them with some examples.

The first example is adding 2 distinct points together, taken from "Elliptic Curve Cryptography: a gentle introduction" by Andrea Corbellini at andrea.corbellini.name/2015/05/17 /elliptic-curve-cryptography-a-gentle-introduction/:

```For the elliptic curve given below:
y2 = x3 + ax + b, where (a=-7 and b=10)
Or:
y2 = x3 - 7x + 10

And two given points:
P = (xP, yP) = (1,2)
Q = (xQ, yQ) = (3,4)

Find the sum of P and Q:
R = P + Q = (xR, yR)

From equation (10):
yP - yQ
m = ---------                (10)
xP - xQ

We get:
m = -2/-2 = 1

From equations (8) and (9):
xR = m2 - xP - xQ              (8)
yR = m(xP - xR) - yP           (9)

We get:
xR = 1*1 - 1 - 3 = -3
yR = 1*(1 + 3) - 2 = 2

So:
R = (-3,2)
```

Here is how we can verify the result:

```Point, -R=(-3,-2), must be on the elliptic curve, which can be verified as:
y2 = x3 - 7x + 10
Or:
(-2)*(-2) = (-3)*(-3)*(-3) - 7*(-3) + 10
4 = -27 + 21 + 10
4 = 4

Point, -R=(-3,-2), must be on the straight line passing through P and Q,
which can be verified as:
y = m(x - xP) + yP
Or:
-2 = m(-3 - 1) + 2
-2 = 1*(-4) + 2
-2 = -2
```