Proportional vs Nonproportional Relationships

You're comparing two cell phone plans. One charges $10 per gigabyte of data. The other charges $5 per gigabyte plus a $20 monthly base fee.

Both look like straight lines on a graph. Both charge per gigabyte. But one is proportional and the other is not. The difference? That $20 base fee.

Here's how to tell them apart — every time, in every format.

The Core Difference in One Sentence

A proportional relationship follows $y = kx$ (no added constant). A nonproportional relationship follows $y = mx + b$ (shifted by $b$).

That $b$ — the constant added or subtracted — is the only thing separating proportional from nonproportional.

Side-by-Side: Tables

Compare these two tables. Both are linear. Only one is proportional.

| $x$ | Proportional $y$ | $y \div x$ | | :-: | :-: | :-: | | 1 | 5 | 5 | | 2 | 10 | 5 | | 3 | 15 | 5 | | 4 | 20 | 5 |

| $x$ | Nonproportional $y$ | $y \div x$ | | :-: | :-: | :-: | | 1 | 7 | 7 | | 2 | 12 | 6 | | 3 | 17 | 5.67 | | 4 | 22 | 5.5 |

Left table: every $y \div x$ equals 5. Constant ratio. $k = 5$. Proportional.

Right table: the ratios change (7, 6, 5.67, 5.5). The $y$ values increase by 5 each time, but they start at 7 instead of 5. This is $y = 5x + 2$. Not proportional.

The test: divide $y$ by $x$. Same every time = proportional. Different = not proportional.

Side-by-Side: Graphs

Both lines below are straight. Both have the same steepness. Look where they cross the $y$-axis.

• Proportional: crosses at $(0, 0)$. Line: $y = 5x$.
• Nonproportional: crosses at $(0, 2)$. Line: $y = 5x + 2$.

$y = 5x \quad \text{vs} \quad y = 5x + 2$

Same slope. Same angle. But that 2-unit shift up changes everything. A line through the origin is proportional. A line shifted off the origin — even by 0.1 — is not.

The test: check $(0, 0)$. If the line goes through it, check ratio consistency. If not, done — not proportional.

Side-by-Side: Equations

| Proportional | Nonproportional | |---|---| | $y = 3x$ | $y = 3x + 4$ | | $y = \frac{1}{2}x$ | $y = \frac{1}{2}x - 3$ | | $y = -2x$ | $y = -2x + 1$ | | $y = x$ | $y = x + 5$ |

Left column: $y = kx$ form only. No $+ b$, no $- b$. Right column: has a constant term that shifts the line.

The test: can you write it as $y = kx$ with nothing added or subtracted? If yes, proportional. If you need $+$ or $-$, it's not.

Why the Origin Matters

The origin rule comes straight from the equation. In $y = kx$:

$y = k(0) = 0$

Plug in $x = 0$, and $y$ must be 0. In $y = mx + b$:

$y = m(0) + b = b$

Plug in $x = 0$, and $y = b$. That $b$ is the shift.

In real terms, $b$ means "something is already there before you start." That's a base fee, a starting charge, an initial distance — anything that exists at zero quantity. Proportional relationships have none of that. Zero of one means zero of the other.

Real-World Examples

Proportional: Gas costs $4 per gallon. Buy 0 gallons = pay $0. Buy 5 gallons = pay $20. Buy 10 = $40. Every gallon adds exactly $4.

$y = 4x$

Nonproportional: A cell plan costs $5 per GB plus a $20 base fee. Use 0 GB = still pay $20. Use 5 GB = pay $45 ($25 data + $20 fee). Use 10 GB = pay $70. The fee doesn't multiply.

$y = 5x + 20$

The fee turns a proportional relationship into a nonproportional one. That $20 is $b$.

Quick Reference

| Feature | Proportional | Nonproportional | |---|---|---| | Equation form | $y = kx$ | $y = mx + b$ ($b \neq 0$) | | Graph | Straight line through $(0, 0)$ | Straight line shifted off origin | | Table ratios | Constant ($y \div x$ same) | Changing ($y \div x$ differs) | | $x = 0$ gives | $y = 0$ | $y = b$ | | Real example | Pay per hour with no base | Pay per hour with sign-up fee |

Practice: Identify Proportional or Not

1. Table:

| $x$ | $y$ | | :-: | :-: | | 2 | 8 | | 5 | 20 | | 7 | 28 |

$8 \div 2 = 4$, $20 \div 5 = 4$, $28 \div 7 = 4$. Constant ratio. Proportional. $k = 4$.

2. Graph: A straight line passes through $(0, 3)$ and $(2, 7)$.

The line doesn't pass through $(0, 0)$. Not proportional. (It's $y = 2x + 3$.)

3. Equation: $y = \frac{3}{4}x - 2$

Has a $- 2$ constant. Not proportional.

4. Word problem: A baker uses 2 cups of flour for every 12 cookies. Write the relationship.

$k = 2 \div 12 = \frac{1}{6}$

$y = \frac{1}{6}x$

No added constant. When the baker makes 0 cookies, they use 0 cups of flour. Proportional.

FAQ

Is a nonproportional relationship still linear? Yes. $y = mx + b$ is linear — it graphs as a straight line. The "nonproportional" label means it doesn't pass through the origin, not that it's curved.

Can a relationship be proportional if $k$ is negative? Yes. $y = -3x$ is proportional. The line goes through the origin and has a constant ratio ($-3$). It just points downward.

What if a table has $x = 0$ and $y = 0$ but ratios differ later? Not proportional. The origin is necessary but not sufficient. You still need every $y \div x$ to match. Check non-zero rows to be sure.

How do I know $b$ from a graph? Look at the $y$-intercept — where the line crosses the $y$-axis. That point is $(0, b)$. If $b \neq 0$, it's nonproportional.

What if the equation is $y = 5x + 0$? That's just $y = 5x$. Proportional. $b = 0$ means no shift. Most teachers won't write the $+ 0$, but if they do, it's still proportional.

Do all proportional relationships have a $y$-intercept of 0? Yes. In $y = kx$, when $x = 0$, $y = 0$. The line always passes through $(0, 0)$. That's the definition.

Can a nonproportional relationship have a constant $y \div x$? No. If $y \div x$ is constant, the relationship is $y = kx$ by definition, which means $b = 0$ and it's proportional.

What's the difference between $k$ and $m$? Nothing, when the line goes through the origin. In $y = kx$, $k$ is the slope. In $y = mx + b$, $m$ is the slope. Same number, same meaning — the rate of change. The only difference is the $b$.

Related: Review the common mistakes students make or get a full breakdown of why the equation is $y = kx$. Use our graph checker, table checker, or equation checker to instantly check any relationship.