Final answer:
After finding the constants of proportionality for each type of variation (joint, inverse, and combined direct and inverse), y equals 63 when x is 81 and z is 7, y equals 5/9 when x is 45, and x equals -10 when y is 9 and z is 5.
Step-by-step explanation:
If y varies jointly as x and z, this can be represented by the equation y = kxz, where k is the constant of proportionality. First, we solve for k using the given values: when y = 12, x = 18, and z = 6. This yields k = y/(xz) = 12/(18*6) = 1/9.
Now, to find y when x = 81 and z = 7, we substitute these values and our k into the original equation:
y = (1/9) * 81 * 7 = 63.
When y varies inversely as x, the equation is y = c/x, where c is the constant of proportionality. Given that y = 5 when x = 5, c = xy = 25. To find y when x = 45, we calculate y = 25/45 = 5/9.
For a scenario where y varies directly as z and inversely as x, the equation is y = (m*z)/x where m is the constant. When y = 27, z = -3, and x = 2, we can find m: y = (m*z)/x, so 27 = (m*(-3))/2 and m = -18. To find x when y = 9 and z = 5, we use our constant m and the equation, resulting in x = (m*z)/y = -18*5/9 = -10.