k is the constant of variation.
you first solve for k and then you use k to solve for the rest of the problem.
example:
y is directly proportional to x.
y = 25 when x = 5
formula is y = k * x
when x = 3, y = 25.
replace y with 25 and x with 5 and solve for k.
y = k * x becomes:
25 = k * 5
solve for k to get:
k = 25/5 = 5.
when k = 5, 25 = k * 5 becomes 25 = 5 * 5 which becomes 25 = 25.
this confirms the value of k is good.
what is the value of y when x becomes 30.
the value of k is used to help solve this.
y = k * x becomes y = 5 * 30 which becomes y = 150.
note that 25 is equal to 5 * 5 and 150 is equal to 5 * 30.
that's direct variation.
as x increases, y increases by the same proportion.
y = 5 * x
x = 5, y = 25
x = 30, y = 150
to graph -2y = 5x, you have to solve for y.
divide both sides of the equation by -2 to get:
y = 5x / -2.
this can be shown as:
y = -5/2 * x
that equations is now in slope intercept form of y = mx + b
m is the slope
b is the y-intercept.
the slope is equal to -5/2
the y-intercept is 0.
the graph of that equation looks like this.
in the graph you can see two points plotted.
points are plotted in (x,y) format.
(0,0) says the value of y is 0 when the value of x is 0.
that's the y-intercept.
(10,-25) says the value of y is -25 when the value of x is 10.
from the equation of y = -5/2 * x, then x = 10, y = -5/2 * 10 which makes y = -25.
the slope is the change in the value of y divided by the corresponding change in the value of x.
the starting point was (0,0) and the ending point was (10,-25)
x went from 0 to 10.
y went from 0 to -25.
the slope is the change in y divided by the corresponding change in x.
-25/10 = -5/2 which is the slope of the equation.
here's some references that might help.
https://www.purplemath.com/modules/variatn.htm
https://www.purplemath.com/modules/strtlneq.htm