Taking a simpler line than your problem, consider the equation y = x.
This line goes through the points (1,1), (3,3), (-2.7, -2.7), etc.
The "equation of a line" is a description of what paired values of x and y describe the set of points that are on the line. An equation is a statement that can be either true or false.
y = x is TRUE when x = 3 and y = 3.
It is false when x = 3 and y = 4.
Therefore the point (3,3) is on the line y=x, and the point (3,4) is NOT on the line y=x.
Now consider the line 2y = 2x.
You'll find that it describes the exact same set of points. This equation statement is true at all the same (x,y) values where y=x is true, and false at all the same values where y=x is false.
In general, multiplying both sides of an equation by the same number (except 0) will produce a different equation that has the same truth values.
y=x is the same as 2y=2x is the same as 17y=17x.
Adding the same number to both sides of an equation, or pretty much any mathematical manipulation done to both sides, produces an equation that describes the same line.
y = x-1 is the same as y+1 = x is the same as y - 3 = x - 4.
This is an important principle for solving algebraic equations.