Determine the equation of joint variation. Then solve for the missing value.

x varies directly with y and z.
x = 400 when y = 8 and z = 10.
Find x when y = 10 and z = 12.

If

"x varies directly with y and z",
that means
x=kyz, where k is a constant.

Since
x = 400 when y = 8 and z = 10
or
400=k(8)(10),
we can readily solve for k=5.

Hence solve for x when
when y = 10 and z = 12
(hint: k=5)

Determine the equation of combined variation. Then solve for the missing value.

x varies directly with y and inversely with z.
x = 5 when y = 10 and z = 5.
Find x when y = 20 and z = 10.

To determine the equation of joint variation, we use the formula:

x = k * y * z

where k is the constant of variation.

To find the value of k, we substitute the given values of x, y, and z into the equation and solve for k:

400 = k * 8 * 10
400 = 80k
k = 400/80
k = 5

Now that we know the value of k, we can find x when y = 10 and z = 12:

x = 5 * 10 * 12
x = 600

Therefore, when y = 10 and z = 12, x = 600.

To determine the equation of joint variation, we know that x varies directly with both y and z. This can be represented as:

x = k * y * z,

where k is the constant of variation.

To find the value of k, we can use the given information that x = 400 when y = 8 and z = 10. Substituting these values into the equation, we get:

400 = k * 8 * 10.

Simplifying, we have:

400 = 80k.

Dividing both sides by 80, we find:

k = 5.

Now we can substitute the value of k back into the equation of joint variation:

x = 5 * y * z.

To find the value of x when y = 10 and z = 12, we substitute these values into the equation:

x = 5 * 10 * 12.

Simplifying, we get:

x = 600.

Therefore, when y = 10 and z = 12, x is equal to 600.