Direct variation is one of 2 variation formulas in the algebra section of this library, and it is used at middle school · high school level.
Why direct variation works
Saying y varies directly with x means the ratio of y to x never changes. Name that fixed ratio k and multiply both sides by x, and you get a line through the origin: doubling the input doubles the output, and zero input gives zero output.
What each symbol means
$k=y/x$ is the constant of variation.
Direct variation: when it holds
$x\ne0$ when calculating $k$; the graph passes through the origin.
When it stops applying
Any relationship whose graph misses the origin is not direct variation. Fahrenheit is 1.8 times Celsius plus 32, so 10 degrees Celsius is 50 Fahrenheit but 20 Celsius is 68, not 100. Doubling one temperature does not double the other.
Direct variation: a worked example
If $y=12$ when $x=3$, then $k=4$ and $y=4x$.
The mistake to avoid
What people do: Calling something direct variation when the rule has a constant added, such as y = 2x + 3.
Why it goes wrong: That relationship has a changing ratio: at x = 1 the ratio is 5, and at x = 2 it is 3.5. A constant term lifts the graph off the origin and breaks proportionality.
Do this instead: Test two data pairs by dividing y by x. If the two quotients differ, the relationship is not direct variation and k does not exist.
Direct variation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $k=y/x$ is the constant of variation.
- Check the conditions before substituting. $x\ne0$ when calculating $k$; the graph passes through the origin.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Variation
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where direct variation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Ratios and Proportions — the lesson behind this formula: compare quantities and scale equivalent relationships.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about direct variation
How do I find k from a single pair of values?
Divide y by x once, since one point is enough for a line through the origin. If y is 12 when x is 3, then k is 4 and the rule is y = 4x.
What does it mean if y varies directly with the square of x?
The rule becomes y = kx squared, so the constant ratio is between y and x squared. Tripling x then multiplies y by nine rather than by three.
Are direct variation and proportional the same idea?
Yes, they are two names for the same relationship. Proportional is the word used in science and everyday contexts, while direct variation is the phrase most algebra textbooks use.
Can I find a new y without computing k first?
Yes, set the two ratios equal and cross-multiply. If y is 12 at x = 3, then at x = 7 the proportion gives y = 28 in a single step.