4x+3y=21

What is your question?

You have simply stated the equation of a straight line.

what is a prime factorazation?>

expressing a number in terms of its Prime factors
eg.

126 = 2*3*3*7

all the factors are prime numbers

How does this relate to the first problem you posted?

The prime factorization of a number is a way to express that number as a product of its prime factors. It doesn't directly relate to the equation you posted. The equation 4x + 3y = 21 is a linear equation, which is a completely different concept in algebra.