How do you write an equation of a line that is parallel to `y+3x=7` and passes through point `(7,2)`?

I would use first your equation to find the SLOPE of your line. Basically the slope is a number that tells you what the inclination of your line is.
So, to find the parallel to your line you need a line with the same inclination...the same slope:
Your line: `y+3x=7` can be written (isolating the `y` on the left) as:
`y=-3x+7` this allows you to "read" immediately the slope of your line, the coefficient of `x`, which in your case is `-3`.

Now the difficult bit...
The slope represents the inclination of your line and basically tells you how `y` changes when `x` changes!

For example, a big slope means that at every fixed change in `x` the value of `y` changes a lot and your line is VERY steep!!!
Have a look at this picture:

slope `5` is steeper than slope `2`!

To find your slope you simply take the change in `y` divided by the change in `x`:

In your case:

`slope=(Deltay)/(Deltax)=(y_2-y_1)/(x_2-x_1)` but you want a slope which is equal to the one of your original line: `-3`
Together with the coordinates of your point you can write:
`slope=(y_2-y_1)/(x_2-x_1)`
`-3=(y-2)/(x-7)`
`-3x+21=y-2` and finally your line:
`y+3x=23`
Graphically:

hope it helps