Direct variation says y = kx: one quantity is a fixed multiple of another, and the multiple never changes. Cost against weight at a fixed price, distance against time at a fixed speed, circumference against diameter. Find the constant from one pair of values and every other pair follows. The condit...
y = 2x + 3 is a straight line and is not a direct variation. Direct variation says y = kx, which forces the line through the origin: no x means no y. Add a constant and doubling x no longer doubles y, so the whole method fails while the graph still looks perfectly respectable. This page tests your data rather than assuming it — enter several points and it will tell you whether one constant really fits them all, and if not, whether they are at least collinear and by how much the line misses the origin.
One point gives a constant. Several points are what test whether it means anything.
X
Y
THE EQUATION
y = 3x
All 3 points give the same constant. That agreement is the evidence; one point alone would have produced a constant regardless.
IF X IS…
y = 30
IF Y IS…
—
What the power means. multiply x by 2 and y goes up by ×2; multiply x by 3 and y goes up by ×3; multiply x by 10 and y goes up by ×10. With a plain x the two factors match, which is the only case where they do.
THE CONSTANT
3
exact
POINTS AGREEING
3 of 3
the relationship holds
AS A DECIMAL
3
exact
THROUGH THE ORIGIN
yes
as direct variation requires
WHAT IS HAPPENING
k = y ÷ x
Every point must give the same k. That is a stronger claim than lying on a straight line, because the line also has to pass through the origin — at x = 0 there must be no y at all.
(2, 6) → k = 3 · (5, 15) → k = 3 · (7, 21) → k = 3
Every point gives the same constant, so the data really does vary directly. One point would have produced a constant too — agreement across several is what makes it evidence.
y = 3x
The constant is exact, so this equation reproduces every point you gave it precisely.
Why the origin decides it. Any two points can be joined by a straight line, so finding a line proves nothing on its own. Direct variation additionally demands that the line pass through the origin, and that is a real restriction: of all the point pairs on a small grid, under one per cent satisfy it. Given a pair that does fit, a third point drawn at random agrees only 4.3% of the time — which is exactly why a third point is worth having.
Exact fractions throughout · consistency tested without ever taking a root
Choose what y varies directly as — plain x, its square, its cube, or its square root. The choice changes what stays constant and what happens when x is scaled.
Enter your data points, one pair per row. One point is enough to produce a constant, but it is agreement across several that tells you the relationship is real rather than a coincidence.
Read the verdict. If every point gives the same constant you get the equation; if not, the page names the first point that disagrees and, where the points are collinear, gives the line and the intercept that disqualified it.
Use the prediction boxes to go either way once the fit holds — supply an x to get y, or a y to get x. Answers stay exact fractions unless a root genuinely makes them irrational, in which case they are marked as approximations.
Start with three points: (2, 6), (5, 15) and (7, 21). Work out k = y ÷ x for each. 6 ÷ 2 = 3, 15 ÷ 5 = 3, 21 ÷ 7 = 3. All three agree, so the relationship is y = 3x. Any further pair follows immediately: at x = 10, y = 30; at y = 45, x = 15. Now three points that look just as tidy: (1, 5), (2, 7) and (3, 9). The constants are 5, 3.5 and 3 — they do not agree, so this is not a direct variation. But the points are not random either. The slope between any two of them is 2, and the line through them is y = 2x + 3. Every point sits on that line, so the relationship is linear; it simply misses the origin by 3. At x = 0 it predicts 3, and direct variation requires nothing at all. This is the single most common false positive in the subject: a set of points that plainly lie on a line, mistaken for a proportional relationship. Try the square. Points (1, 5), (2, 20) and (3, 45) give constants of 5 ÷ 1, 20 ÷ 4 and 45 ÷ 9 — all 5 — so y = 5x². Note what this means for scaling: doubling x from 2 to 4 does not double y from 20 to 40, it quadruples it to 80. The factor applied to y is the factor applied to x raised to the power, and it equals that factor only when the power is 1. The square root runs the other way. Points (4, 6), (9, 9) and (16, 12) give 6 ÷ 2, 9 ÷ 3 and 12 ÷ 4 — all 3 — so y = 3√x. Quadrupling x doubles y. Getting an exact 3 here rather than 2.9999999996 matters: the constants are compared as k² = y² ÷ x, which stays a rational number and never requires a root to be taken at all. Finally a case where the constant is genuinely irrational: (2, 2), (8, 4) and (18, 6). Each gives k² = 4 ÷ 2, 16 ÷ 8, 36 ÷ 18 — all exactly 2. The points agree perfectly, so the relationship holds, but k is √2 and no decimal will ever write it out. The honest answer is that k² = 2 exactly and k ≈ 1.414214, and that is what gets shown.
| Rule | What it says | Why |
|---|---|---|
| Direct variation | y = kx | y varies directly as x. One constant, and the line goes through the origin. |
| The constant | k = y ÷ x | Every point must give the same k. That is the whole test. |
| Through the origin | x = 0 forces y = 0 | The defining condition. A line missing the origin is not a direct variation. |
| Linear is not enough | y = 2x + 3 fails | Perfectly straight, and doubling x does not double y. The + 3 spoils it. |
| Varies as the square | y = kx² | Doubling x multiplies y by 4. Area against radius behaves this way. |
| Varies as the cube | y = kx³ | Doubling x multiplies y by 8. Volume against length behaves this way. |
| Varies as the root | y = k√x | Quadrupling x doubles y. Pendulum period against length behaves this way. |
| The scaling factor | multiply x by f, y by fⁿ | Never by f unless n is 1. This is what the power actually means. |
| Two points prove nothing | a third is the test | Given a pair that fits, a random third point agrees only 4.3% of the time. |
| Inverse variation | y = k ÷ x | The opposite: x × y stays constant. A different page and a different shape. |
| Joint variation | y = kxz | Directly as two things at once. Same idea, one more variable. |
| Finding k | one point is enough | One point always gives a k. It is agreement across points that means anything. |
| Units | k carries them | If y is cost and x is kilograms, k is cost per kilogram, and it is not dimensionless. |
| The graph | a curve through the origin | Straight when n = 1, bending up when n > 1, bending down when n < 1. |
It is the relationship y = kx, where k is a fixed number called the constant of variation. Doubling x doubles y, tripling x triples y, and the graph is a straight line through the origin. "y varies directly as x", "y is directly proportional to x" and "y = kx" all say the same thing. The constant carries the units of the relationship — if y is cost and x is weight, k is a price per unit weight.
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Last updated: August 6, 2026 · Exact fractions throughout, never floating point · Direct variation requires the line to pass through the origin.