Homogeneity and Ratio Substitution

A homogeneous expression is one in which every term has the same total degree.

For example,

is homogeneous of degree 3, since every term has total degree 3.

We can often divide by a suitable power of one variable and introduce ratios such as

This can decrease the number of variables and turn a complicated equation or inequality into a simpler one.

Key Idea

When all terms have the same total degree:
1. Divide by a suitable power of one nonzero variable
2. Replace ratios of variables by new variables
3. Solve the simpler problem involving fewer variables