# A naive question about the Fredholm equation

I am new to integral equations. In this field, people study the Fredholm equation

$$\phi(x) + \int_0^1 K(x, y) \phi(y) dy = f(x).$$

I am a bit surprised to see the first term on the left hand side. In linear algebra, we have the equation

$$\sum_j A_{ij} x_j = b_i .$$

Here $$A$$ is the counterpart of $$K$$, and $$b$$ the counterpart of $$f$$. Therefore, the simplest and most natural integral equation for me would be

$$\int_0^1 K(x, y) \phi(y) dy = f(x).$$

So why did people not start from this one, but the strange one above? This might be a very naive question, but I did not find any discussion in the literature.