Direct Variation: The Equation y = kx, How to Find k, and Which Tables and Graphs Qualify

§ 01

What is direct variation?

What is direct variation? The direct variation meaning is narrower than it sounds: it's not just "two things that go up together," it's specifically a relationship where the ratio between them, $y/x$, stays exactly constant at every point — a stricter test than just both variables trending the same direction. The direct variation definition rules out anything with an added constant, anything curved, and anything where the ratio drifts even slightly as the numbers change. Once you have a candidate value of $k$, confirming it works is checking a value by substitution applied to this specific equation form — plug a known $x$ and $y$ pair back into $y=kx$ and see whether both sides match.

§ 02

The direct variation equation: y = kx

The form y = kx and the constant of variation

What is direct variation in math? It's a relationship where $y$ is always the same multiple of $x$: $y=kx$, where $k$ is the constant of variation and $k \neq 0$. Double $x$ and $y$ doubles too; the ratio $y/x$ never changes. This is what the direct variation equation looks like — no constant term, ever, because the graph must pass through the origin.

Every direct variation problem reduces to finding this one number, $k$, once the form itself is confirmed — after that, the formula is completely determined.

§ 03

How to find k from a table, a point or a graph

Finding k from a table or a point

Scroll the table sideways to see every column.

Every ratio $y \div x$ in this table equals 3, so $k=3$ and the direct variation equation here is $y=3x$. From a single point instead — say $(5,20)$ — the same idea applies: $k=20\div5=4$, giving $y=4x$. A mole-ratio proportion in a stoichiometry problem is a direct-variation relationship too, the kind covered in chemistry and biology homework help — the word problems further down this page work through a physics example of the identical pattern.

Which table represents a direct variation function?

Which table represents a direct variation function comes down to the same ratio check: divide $y$ by $x$ in every row, and see whether the result is identical every time. The table above passes — every ratio equals 3. A table where the ratio changes row to row fails, no matter how orderly the numbers otherwise look, because a changing ratio means the relationship isn't a constant multiple after all. This check works on any number of rows — two rows are enough to rule a table out the moment their ratios disagree, but confirming a table qualifies means checking every row, since one mismatched pair anywhere in the table is enough to fail it.

Finding k from a graph

A direct-variation graph is a straight line, so $k$ is just that line's slope: rise over run. A line that climbs 3 units for every 1 unit it runs has slope 3, so $k=3$ and the equation is $y=3x$ — the identical line the table above describes. Reading $k$ off a graph and reading $k$ off a table or a point are three routes to the same number, not three different numbers.

§ 04

Which equation represents a direct variation?

Which equation represents a direct variation comes down to one test: does it fit the form $y=kx$, with nothing added or subtracted?

Which equation represents a direct variation - spotting the impostors (y = kx + b is not one)

Which of these represents a direct variation?

• A) $y=4x$ — yes. Pure multiple of $x$, $k=4$, passes through the origin.
• B) $y=4x+2$ — no. The added constant means the graph misses the origin; this is $y=kx+b$, a different family entirely.
• C) $y=4/x$ — no. This is inverse variation: the product $xy$ stays constant, not the ratio.
• D) $y=x^2$ — no. It isn't linear at all, so the ratio $y/x$ changes as $x$ changes.

Under Common Core 8.EE, this is exactly the distinction eighth-grade standards test: a direct variation equation, not any equation that happens to contain both $x$ and $y$.

§ 05

Direct variation graphs: a straight line through the origin

Every direct variation graph passes through the origin, $(0,0)$, because $y=k(0)=0$ regardless of what $k$ turns out to be — there's no way to write a direct-variation equation that misses that one point.

Scroll the table sideways to see every column.

The line passes through $(0,0)$ and climbs at a constant rate on both sides of it. Compare that with $y=3x+2$: at $x=0$, $y=2$, not 0 — the graph shifts up off the origin, which is the fastest visual test for ruling a candidate equation out before doing any algebra on it at all. A negative $k$ still produces a straight line through the origin; it just slopes downward instead of upward, falling from left to right instead of rising.

§ 06

Find a direct variation model that relates y and x

Finding a direct variation model that relates y and x is the same three-step process every time: find $k$ from whatever information is given, write the equation, then use it to predict a new value.

A spring: Hooke's law

A spring stretches 6 cm when a 2 kg mass hangs from it. Stretch distance $d$ varies directly with mass $m$: $d=km$. From the given values, $k=6\div2=3$, so the model is $d=3m$. Predict the stretch for a 5 kg mass: $d=3(5)=15$ cm. This is Hooke's law and other physics relationships in its simplest form — force or stretch as a constant multiple of what's causing it.

An hourly wage

Earnings $E$ vary directly with hours worked $h$. A paycheck shows $90 earned for 12 hours: $k=90\div12=7.5$, so $E=7.5h$. Predict earnings for a 20-hour week: $E=$$7.5(20)=$$E=7.5(20)=150$.

Distance at a constant speed

Distance $d$ varies directly with time $t$ at a constant speed. A car covers 150 miles in 3 hours: $k=150\div3=50$, so $d=50t$, and speed itself is the constant of variation here. Predict the distance after 5 hours: $d=50(5)=250$ miles. Notice the pattern across all three examples: $k$ is always found the same way, by dividing a known output by its matching input, regardless of what the two quantities actually represent.

§ 07

Direct vs inverse variation, and y = kx²

In direct variation, y = kx: the ratio y/x stays fixed, and for positive k, y rises as x rises. In inverse variation, y = k/x: the product xy stays fixed, and y falls as x rises.

A third relative, $y=kx^2$, is direct variation with the square instead of $x$ itself — for positive $k$ and $x$, $y$ still grows as $x$ grows, but faster, since doubling $x$ quadruples $y$ instead of doubling it. A dropped object's fall distance varies directly with the square of the time it has been falling: $d=16t^2$ (feet, seconds). After 3 seconds, $d=$$16(3{)}^2=$$16(9)=$$d=16(3)^2=16(9)=144$ feet. Solving for $t$ when $d$ is given instead turns this equation into a quadratic — see solving the quadratic that y = kx² can produce for that method.

§ 08

Practice: ten items with answers

1. Is $y=7x$ direct variation? — Yes, $k=7$.
2. Is $y=7x+2$ direct variation? — No, it has a constant term.
3. Is $y=x/5$ direct variation? — Yes, equivalent to $y=(1/5)x$, so $k=1/5$.
4. Table $x$: 1, 2, 3; $y$: 4, 8, 12 — direct variation? — Yes, $k=4$ (every ratio matches).
5. Table $x$: 1, 2, 3; $y$: 5, 8, 11 — direct variation? — No, the ratios are 5, 4 and 3.67 — not constant.
6. If $y=9$ when $x=3$, find $k$ and write the equation. — $k=3$, so $y=3x$.
7. Using $y=3x$ from item 6, find $y$ when $x=10$. — $y=30$.
8. Is $y=2/x$ direct variation? — No, that's inverse variation.
9. A recipe uses 2 cups of flour per 12 cookies. Write a direct variation equation relating cups $c$ to cookies $n$, and find the cups needed for 30 cookies. — $k=2\div12=1/6$, so $c=n/6$; for $n=30$, $c=5$ cups.
10. Does the graph of $y=-4x$ pass through the origin? — Yes — every direct-variation graph does, regardless of the sign of $k$.

§ 09

When the algebra unit is due

Spotting direct variation and finding $k$ takes a minute once the one test — does it fit $y=kx$ — becomes automatic. A full algebra problem set covering direct variation alongside equations, inequalities and word problems is a longer job, and it's where most students look for homework help across math and science instead of working every item alone. GradeDraft's math desk sends back an algebra set worked step by step, priced from $25 per problem after a quick look at the set.