Most bodies can be decomposed into smaller pieces, in which higher order effects can be ignored. Sometimes each element has to be made up of two pieces. For example,
When solving differential equations, a trick that can be used when the final equation has no dependence on $t$ is to rewrite $a = v \,\mathrm{d}v/\mathrm{d}x$ so the equation of motion is only in terms of $x$. This is generally useful when a dependence on $t$ is not needed (factors of $\textrm{d}t$ can be eliminated through multiplying by $v=\mathrm{d}x/\mathrm{d}t$)
Envelope of a projectile fired at speed $v_0$
$$ y=\frac{v_0^2}{2g}-\frac{gx^2}{2v_0^2} $$
A particle sliding down a string can reach a spherical volume of diameter $\frac{1}{2}gt^2$ in time $t$
The maximum number of linearly independent equations (for force and torque balance) is equal to the number of degrees of freedom of the body.
Hence, to solve a 2d statics problem, you need 2 force balance equations (1 for each axis, not necessarily perpendicular) and 1 torque balance equation, since rigid bodies have 3 degrees of freedom in 2d (assuming there are no constraints which limit the degrees of freedom). Additional torque balance can be done in place of a force balance.
If there are 3 forces acting on an object in equilibrium (in 2D), they are either parallel or concurrent.
If the forces are concurrent (converge at a common center), then if there is zero net force, there is also zero net torque (as it implies zero torque about the common center).
If the forces are parallel, the net force in one axis is already zero.
Once these directional constraints are met, only 2 quantities can be solved for since there are 2 independent equations to solve.
Is it possible to combine two forces such that they exert the same net force and torque everywhere? See this: https://en.wikipedia.org/wiki/Centers_of_gravity_in_non-uniform_fields
Static friction is very notorious since it doesnt have a fixed value. A good way to think of it is to combine it with the normal force to produce a contact force that has a maximum angle. Then we can combine this with the methods mentioned earlier to solve the system geometrically.
Some more tips: