The equation of the line parallel to the given line that contains point $ A $ is:
$ \color{blue}{ x-y-1=0 }$ ( General form )
$ \color{blue}{ y = x - 1 } ~~~$ ( Slope y-intercept form )
Step 1:The slope of a given line is $ m = 1 $.
Step 2: Parallel lines have the same slope, so the slope of the unknown line ($ m_1 $) will also be $ 1 $. So the parallel line will have a slope of $ m_1 = 1 $
Step 3: Now we have a point and the slope so we can use point-slope form, which is:
$$ y - y_0 = m_1 (x - x_0) $$In this example we have: $ m_1 = 1 $ , $ x_0 = 1 $ and $ y_0 = 0 $. After substitution we have:
$$ \begin{aligned} y - y_0 =& ~ m_1 (x - x_0) \\ y - 0 =& ~ 1 ( x - 1) \\y =& ~ x - 1\\ \end{aligned} $$