How do I use the conservation of linear momentum to solve problems?

The linear momentum of a particle is defined as the product between the particle’s velocity (a vector) and the particle’s mass (a scalar quantity):
p=m·v.
The law of conservation of momentum states that the total amount of momentum of a system (that is, the vector sum of the linear momenta of each of the bodies that make up the system) remains constant throughout any physical interaction that the system may undergo, so that the total momentum before the interaction equals the total momentum after the interaction:
pinitial = pfinal
We can now use this law to calculate the velocities (or masses) of bodies before and after physical interactions.
For instance, consider a collision between two particles A and B of masses mA and mB, respectively. For simplicity, let A be travelling from left to right on the x-axis with initial velocity vA0, and B is travelling from right to left with velocity -vB0. We are told that A and B collide, and after the collision particle A travels backwards with velocity -vA1. What is the velocity of B after the collision?
To use the conservation of linear momentum to solve this problem, we begin by writing the expression for the total momentum before the collision and after the collision:
pi=mA·vA0 – mB·vB0 (Note the minus sign before the momentum of B – this is because particle B is travelling in opposite direction to particle A).
pf=-mA·vA1+mB·vB1
We now write down the law of conservation of momentum,
pi = pf
and substitute in our expressions for the momenta:
mA·vA0 – mB·vB0=-mA·vA1+mB·vB1.
Finally, since we are looking for the velocity of B after the collision, we simply need to solve this equation for vB1:
vB1=(mA·vA0-mB·vB0+mA·vA1)/mB.
This general scheme can be expanded to problems of more than 1 dimension. All we must do in such cases is equate the system's initial and final momenta in every different direction, and solve these equations to find the component of the required velocity in the corresponding direction.
Eg. for two dimensions (x, y), the law of conservation of momentum is written as:pinitial = pfinal;x-direction: pi(x)=pf(x)y-direction: pi(y)=pf(y)