Question 1133871
the formula for inverse variation is y = k/x.


solve for k and you get k = xy.


k is the constant of variation.


it always remains the same.


what this means is, if you multiply x by some factor, you need to divide y by the same factor.


for example, let x = 4 and y = 8


the constant of variation is 4 * 8 = 32


if you multiply 4 by 2, then you have to divide 8 by 2.


you will get 8 * 4 = 32.


k has to stay the same.


that is what is meant by the constant of variation.


the formula for direct variation is y = k * x


solve for k to get k = y / x.


if you multiply y by some factor, you also have to multiply x by the same factor in order to keep k the same.


for example, let x = 4 and y = 8.


y / x = 2 = k


if you multiply y by 2, you have to multiply x by 2 as well in order to keep k constant.


you will get 16 / 8 = 2.


the same concept applies whether or not the values involved are positive or minus.


for example:


when x = -2, y = -6


k = -2 * -6 = 12


that's your constant of variation.


if x doubles, then y must be cut in half.


so when x becomes -4, y must becomes -3.


k = xy becomes -4 * -4 = 12.


k is constant.


what happens when x becomes 4.


4 * y = 12 in order to keep k constant.


solve for y to get y = 3.


when x = -2, y = -6
when x = -4, y = -3
when x = 4, y = 3


the constant of variation was always 12.
x * y = k
if x is multiplied by a factor, y must be divided by the same factor in order to keep k constant.


this can be seen graphically below.


<img src = "http://theo.x10hosting.com/2019/020201.jpg" alt="$$$" >


your problem states:


if y = -6 when x = -2, find y when x = 5.


the formula is the same.


y = k / x


solve for k to get k = y * x.


when y = -6 and x = -2, you get k = 12


what happens when x = 5?


x * y = 12


when x = 5, the formula becomes 5 * y = 12


solve for y to get y = 12/5 = 2.4


-2 * -6 = 12
5 * 2.4 = 12


the constant of variation is always k.


you multiplied -2 by -2.5 to get 5.
you divided -6 by -2.5 to get 2.4.


inverse variation says that whatever you multiply x1 by to get x2, you need to divide y1 by to get y2.


in your problem, x1 was -2 and y1 was -6 while x2 was 5 and y2 was 2.4


you multiplied x1 by -2.5 to get x2 (-2 * -2.5 = 5)
you divided y1 by -2.5 to get y2 (-6 / -2.5 = 2.4)


this can be seen graphically below.


<img src = "http://theo.x10hosting.com/2019/020202.jpg" alt="$$$" >