Key here is to distinguish between the general equation of a line, the point slope formula, and the slope intercept equation of the line.
Since you are looking to find the slope-intercept equation of a line, I will cover that first.
The slope intercept form is:
y = mx + b,
Slope intercept form is useful for finding the slope and y intercept of a line, hence why it is called "slope intercept" because it is easy to see the slope and the y intercept of a line in this form. Since the slope denoted by m, and y intercept denoted by b are clearly given.
To get from any linear equation given whether it is a general equation of a line, or the point slope form, just solve for y (in other words get y by itself). So from where you are starting from this means..
y - 3 = 2 (x - 1)
Distribute 2 across the parenteses by multiplying x and -1 by 2.
y - 3 = 2x - 2
add 3 to both sides. This cancels the 3 on the left and moves it to the right, while keeping the equation balanced.
y - 3 + 3 = 2x - 2 + 3
y + 0 = 2x + 1
y = 2x + 1
As you can see our y = 2x + 1 now looks like the point slope equation we had above. m = 2, and b = 1. This means the line goes up 2 for every single unit you move to the right, and intersects the y axis at (0, 1).
The general equation of a line is as follows:
Ax + By = C,
where A, B, and C are constants. This form is very useful for quickly finding x, and y intercepts. Just set x = 0 to find the y intercept, and set y = 0 to find the x intercept. The drawback of the general equation of a line is that the slope andy y intercept isn't infront of you.
The point slope intercept equation is as follows:
(y2 - y1) = m (x2 - x1)
Point slope is literally an addaption of the slope formula, and is useful for finding the equation of a line given a single point, and the slope of the line, hence why it is called "point slope". To find the equation of the line using point slope simply plug in the given points as x1, y1, and m. Plug in x as your x2, y as your y2 and then solve for y. You now have the equation of the line in the slope intercept form thanks to the point slope formula.