I under stand most of it in the testbook section. At least until it gets to the solutions:
L1 = 5 A
L2 = 4A
L3 = -1A
http://www.allaboutcircuits.com/vol_1/chpt_10/2.html
Could some one please tell me what type of algebra this is so I can learn it and understand the branch current method

 

The key point is:

Kirchhoff's Current Law (KCL) tells us that the algebraic sum of currents entering and exiting a node must equal zero.

After the equations for the currents are written out, you have three equations in three unknowns. You likely slept through the math class about how to solve the equations. Solving those equations is simple algebra.

 

Could some one please tell me what type of algebra this is so I can learn it and understand the branch current method

Formally the type of maths used is called 'simultaneous equations', and this is what you you look up in your maths book.

Simultaneous means that all of the equations (three in this case) must be satisfied together.
So that the value of I1 must be the same in each equation (this is true for I2 and I3).

It means that each of Ohm's law, Kirchoff's current law and Kirchoff's law must be satisfied.

Maths note
You require exactly as many equations as there are unknown variables to find a solution.
In this case that is 3

Maths aside some electric circuit theory to note.

1) Kirchoff's current law only has currents as unknowns, but there is only one such equation in this circuit (ie not enough to find a solution.)
This is also called the node current method.

2) Kirchoff's voltage law also uses these currents as variables KVL can be written in terms of voltages and this is useful when you need to find an unknown voltage.

The AAC Ebook has laid it out simply using the basic laws only.

The names mesh current method and branch current method are sometimes used to refer to short cut methods which combine KCL and KVL and many here will use these.

Final point
Thank you for including a proper link to the page your are asking about.
It is a pleasure to not have to try and play detective as to which part of the Ebook your questions refers.

 

It is called linear algebra or simultaneous equations. There are some websites that if you enter in a set of equations it will solve them for you. http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=sys

This can be very useful. Set up the problem by hand, and then solve the equations with the website. Keeps the focus on the concepts as opposed to arithmetic.