An example of simultaneous equations.

In an atom, there is proton and a neutron.

In protons and neutrons, there are quarks. There are different kinds of quarks, one up and one down. These carry the electrical charge.

A proton has a charge of +1.
A neutron has a charge of 0.

In a proton there are two up quarks, and one down quark.
In a neutron there are two down quarks, and one up quark.

By using simultaneous equations, one can find the charge of each quark.

2u + d = 1 (proton)
u + 2d = 0 (neutron)

double the neutron equation, to get

2u + 4d = 0
take the other equation...

2u + 4d = 0
-2u + d = 1

3d = -1
d = -1/3

so a down quark has a negative 1/3 charge.

Take this value and put it in an equation...

2u + d = 1
2u + - 1/3 = 1
2u = 1 and 1/3 = 4/3
u = 4/3 divided by 2.
u = 2/3

So an up quark has a positive 2/3 charge.

up = 2/3
down = -1/3

You can substitute these values to test the validity.