The equation y = x - 12 is only there to give you the slope. To write an equation of a line that is parallel to the given equation, the slopes have to be the same. (If the equation of a line is PERPENDICULAR to the given equation, the slope would be the opposite. For example, if the slope is positive 4, the new slope would be -1/4. If the slope is -2, the new slope is 1/2.)
You are using the equation y = mx + b
where m = slope, b = y-intercept
Since you know the slope of the given equation is just 1, you plug in 1 for m.
y = 1x + b or y = x + b
With the point P(10,3), you just plug it in to y = x + b to find "b" or the y-intercept.
3 = 10 + b
-7 = b
Your new equation is y = x - 7
