The problem states that a straight line $l$ passes through the point $P(1, 3)$ and has a slope of $1$. We need to find the equation of the line $l$.

1. Problem Description

The problem states that a straight line $l$ passes through the point $P(1, 3)$ and has a slope of $1$. We need to find the equation of the line $l$.

2. Solution Steps

We can use the point-slope form of a linear equation:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

In this problem, we are given the point $(1, 3)$ and the slope $m = 1$. Substituting these values into the point-slope form, we get:

$y - 3 = 1(x - 1)$

$y - 3 = x - 1$

$y = x - 1 + 3$

$y = x + 2$

3. Final Answer

The equation of the line $l$ is $y = x + 2$.

So the answer is B.