Question 1186791
direct variation formula is y = k * x


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.


<img src = "http://theo.x10hosting.com/2021/102701.jpg" >


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.


<a href = "https://www.purplemath.com/modules/variatn.htm" target = "_blank">https://www.purplemath.com/modules/variatn.htm</a>


<a href = "https://www.purplemath.com/modules/strtlneq.htm" target = "_blank">https://www.purplemath.com/modules/strtlneq.htm</a>