Variation · 1.3.1
Combined variation
This standard asks students to combine direct and inverse variation into one relationship among three or more variables, for example y varies directly as x and inversely as z. Students must write the correct combined equation y = kx/z, identify the constant k from given data, and use it to find unknown values.
The official learning standard (1.3.1)
“Determine the relation between three or more variables for a combined variation.”
What it means
This standard asks students to combine direct and inverse variation into one relationship among three or more variables, for example y varies directly as x and inversely as z. Students must write the correct combined equation y = kx/z, identify the constant k from given data, and use it to find unknown values.
How it is examined
In SPM 1449 this appears in Paper 1 as multiple-choice items asking students to write the correct combined variation equation from a worded statement, and in Paper 2 as structured questions where students find the constant k, form the equation, then calculate an unknown variable from given values.
Worked example
y varies directly as x² and inversely as z. When x = 2, z = 4, y = 6.
Find y when x = 3, z = 9.
- Write general equation: y = kx²/z.
- Substitute x = 2, z = 4, y = 6: 6 = k(2²)/4 = 4k/4 = k, so k = 6.
- Equation: y = 6x²/z.
- Substitute x = 3, z = 9: y = 6(3²)/9 = 6(9)/9 = 6.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I know if a variable is direct or inverse in a combined variation?
Look for language like "varies directly as" (that variable multiplies, goes on top) and "varies inversely as" (that variable divides, goes on bottom). Combine all such variables with a single constant k in one equation, e.g.
y = kx/z, then use given values to solve for k.
What's the difference between joint variation and combined variation?
Joint variation is a direct variation only, where one variable varies as the product of two or more variables (e.g. y = kxz).
Combined variation mixes direct and inverse relationships together, so it includes at least one variable that varies inversely (e.g. y = kx/z).
Can combined variation involve powers like x²?
Yes. The variable can vary directly or inversely as a power of another variable, such as x² or √x.
The method stays the same, write y = kx²/z (or similar), substitute known values to find k, then use the equation to find the unknown value.