Variation
Joint variation
A relationship where one quantity varies with two or more others at once.
| English | Joint variation |
|---|---|
| Bahasa Melayu | Variasi tercantum |
| 中文 | 联合变化 |
How it is used
If y varies directly as x and the square of z, then y = kxz². Given y = 36 when x = 2 and z = 3, then 36 = k × 2 × 3² = 18k, so k = 2 and y = 2xz².
When x = 4 and z = 2, y = 2 × 4 × 4 = 32.
Where it shows up in SPM
Joint variation is the last and hardest sub-topic of the Form 5 Variation chapter. It appears mainly in Paper 2, linking three quantities, for example y directly as x and inversely as z; you find k, then predict a new value.
Don't confuse it with
Frequently asked questions
How is joint variation different from direct variation?
Direct variation connects y to a single quantity, y = kx. Joint variation connects y to two or more at the same time, such as y = kxz or y = kx²z.
You still find one constant k, but you substitute values for every variable at once.
How do I find k when there are two variables?
Substitute the full set of matching values into the model and solve. For y = kxz² with y = 36, x = 2, z = 3: 36 = k × 2 × 9 = 18k, so k = 2.
Then y = 2xz² answers any later part.
What does 'varies directly as x and inversely as z' mean?
It combines both types into y = kx/z. y rises when x rises and falls when z rises.
Find k from one data set: if y = 10, x = 4, z = 2, then 10 = k × 4 ÷ 2 = 2k, so k = 5 and y = 5x/z.