What is meant by the statement- "a linear Diophantine equation is an equation between two sums of monomials of degree zero or one"?

Question

I know that $ax+by=c$ is a linear Diophantine equation with a,b, and c as constants.

If the aforementioned statement means that one of the monomials can be of degree zero then isn't it is the contradiction to the definition of linear Diophantine equation which demands that the number of variables should be greater than or equals to two?

Also, from the same statement it appears that there can be only two monomials sum in a linear Diophantine equation. Isn't it is possible that the linear Diophantine equations in 3 variables can exist?

PS- the statement in the question was taken from the Wikipedia page and so does the definition of linear Diophantine equations.

Answer 1

Let's separate the terms which make the phrase:

"A linear diofantine equation is an equation between two (sums) of (monomials) of (degree zero) or ([degree] one)"

1. A monomial is some expression of the form $ax^k$, where $x$ is a variable and $a$ is a number (probably, you consider only integers).

2. A monomial of degree zero has the form $ax^0=a$, where $a$ is a number

3. A monomial of degree one has the form $ax^1=ax$, where $x$ is a variable and $a$ is a number.

4. A sum of monomials of degree zero or one will thus be of the form $a_1x_1+a_2x_2+\cdots+a_nx_n+b_1+b_1+\cdots+b_m$< where $x_i$ are variables and $a_i,b_i$ are constants. We can in fact add all the zero-degree monomials, i.e., consider $b=b_1+\cdots+b_m$, so a sum of monomials of degree zero or one will have the form $$a_1x_1+a_2x_2+\cdots+a_nx_n+b$$ where the $x_i$ are variables and the $b,a_i$ are numbers.

5. Thus a linear diofantine equation is an equation between two of the terms above, i.e., of the form $$a_1x_1+\cdots+a_nx_n+b=a_1'x_1+\cdots+a_n'x_n+b'$$ where $x_i$ are variables, $b,b',a_i,a_i'$ are numbers.