Nailed it!

They need to calculate tolerance and don't want to break the bridge or lose a truck. They just keep driving bigger trucks until it starts making a weird sound and then they call the limit at that weight.

The polybridge technique

Depends on how smart you were about sneaking math in as something fun and useful.

If the child somehow got to this point, go ahead. It will spare them a lot of headaches.

For a plank, it is. For anything else, where you are using more complex shapes and more than one material, no finite firm is known, so FEM all the way (or some other simulation technique)

Matrix math is just shorthand for “regular algebra” with a lot of variables

What's pictured here is the Finite Element Method. Basically, the bridge is composed into little chunks, on which the physics solution is approximated by e.g. linear functions. Then one has to assemble and solve a system of equations (the matrix shown here) to determine the coefficients (i.e. slope) of those linear functions. Putting everything together gives a piecewise-linear approximation of the physics solution, and the smaller you make the chunks the better your solution gets. All of this is of course not done by hand, that's what computers are for.