y= -108 when x= -4 and z=3 ; x=6 and z= -2?

Question

Okay this is Algebra 2, and this is Joint Variation(y=kxz) the directions were to write the appropriate joint-variation equation and find y for the given values of x and z. HOW!! The answer in the book says the answer is 9xz ; -108.......but if u can explain in detail to how one comes to that answer =) Thanks in advance!!!

Answer 1

The idea is to first find what's called the constant of joint variation. In this particular case, it's the constant of direct joint variation, because as the variables increase or decrease, the result changes directly in the same manner. Also, if the result has increased or decreased, it implies that the variables have done the same. This constant will tell you how the joint variables relate to each other no matter what they are.

Given y = (k)xz, then the constant of variation, k, = y/xz. Once you have found this constant, you can plug it into the equation given any two variables to find the third. In your example:

-108 = k (-4)(3) = -12k
-108 = -12k
(-108/-12) = k
9 = k

Now just plug 9 in for k in your joint variation equation and you have it.

y = (9) xz

Just plug k = 9 and the other given variable values into the above equation to find y.

y = (9) xz
y = (9) * 6 * -2
y = -108

Once you know k, if you are given any two variable values, you can solve for the third. Suppose y = 144 and x = 8. Manipulating the variation equation algebraically to find z in terms of k, z and y, we get:

z = y / (k)x

Now we plug in the values for k, x and y:

z = 144 / (9) * 8
z = 144 / 72
z = 2

As a check, sub all the variables back into the original equation of variation to see what results:

? = y = (9) * 8 * 2
? = y = 144

So, the answer is correct.

Anyway, that's how joint variation works. If you think this is fun, wait till you get to inverse joint variation:

y = (k) 1/xz, where y increases as the product of x and z decreases or y decreases as the product of x and z increases.

There are also problems like this:

y = (k) x/z, where y varies directly as x and inversely as z.

Isn't math fun? So many possibilities. So many potential headaches.

Answer 2

For the first part, find out what k with the info given to you in the first part of the question.

-108=(-4)(3)k
-108=-12k
9 = k


Next, use k (your constant) to find out the answer for the second euqation
y = (6)(-2)(9)
y = -108

Answer 3

I finished 1-30 but it is in a PDF how should i get it to you?