Value of the y-intercept of the line that passes through the points (5,2) and has a slope of -2/5?

Underscores indicate indices, as in P_1( x_1, y_1) = P_1( 5, 2).

You've got a point P_1( 5, 2) and the slope m = -2/5. So, let's write the equation in the point-slope form y - y_1 = m (x - x_1) :

y - 2 = -2/5 (x - 5)

And now we convert this into the slope-intercept form y = mx + b:

y - 2 = -2/5 (x - 5) ;;; isolate y

y = -2/5 (x - 5) + 2 ;;; expand the term on the right (get rid of parentheses)
y = -(2/5) x + 2 + 2
y = -(2/5) x + 4

The y-intercept is at point P_0( 0 , 4).

Note that b is NOT equal to y_1. It's rather b = m * -x_1 + y_1.