Variation · 1.0.3
Joint variation of three variables
This standard extends direct variation to joint variation, where one variable varies directly as the product of two or more other variables, written as y = kxz (or with more variables). Given one complete set of matching values, you find the constant k, then write the full relation connecting all the variables involved.
The official learning standard (1.0.3)
“Determine the relation between three or more variables for a given joint variation.”
What it means
This standard extends direct variation to joint variation, where one variable varies directly as the product of two or more other variables, written as y = kxz (or with more variables). Given one complete set of matching values, you find the constant k, then write the full relation connecting all the variables involved.
How it is examined
This is mostly tested in Paper 2, as one part of a structured question on variation. You are usually given a full set of values for all the variables to find k, then asked to find an unknown value when some of the variables change to new given values.
Worked example
y varies directly as the product of x and z. When x = 2 and z = 3, y = 24.
Determine the relation between y, x and z, and hence find y when x = 5 and z = 4.
- Since y varies directly as the product of x and z, write y = kxz.
- Substitute x = 2, z = 3, y = 24: 24 = k(2)(3) = 6k, so k = 4.
- The relation is y = 4xz.
- When x = 5, z = 4: y = 4(5)(4) = 80.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What's the difference between joint variation and combined variation?
In joint variation, one variable varies directly as the product of two or more others, so all the relationships are direct (y = kxz). In combined variation, direct and inverse relationships are mixed in the same equation, such as y = kx ÷ z.
How many variables can be involved in a joint variation question?
At least three in total, one dependent variable (y) and two or more that it varies with, like x and z, for example y = kxz or y = kxzw. The method for finding k stays the same no matter how many variables are involved.
What if one of the other variables, like z, stays fixed while x changes?
Then you can combine k and the fixed value of z into a single new constant, say K = kz, so the relation simplifies to y = Kx, an ordinary direct variation between y and x alone.