Why Proportional Relationships Look Like y = kx

Why $y = kx$? Why not $y = kx + b$? Why not $y = \frac{k}{x}$?

The equation $y = kx$ isn't arbitrary. It's the direct result of what a proportional relationship actually is: two quantities that change at a fixed, constant rate.

What Each Part Means

$y = kx$

• $y$ — the dependent quantity (what you're figuring out)
• $k$ — the constant of proportionality (the fixed rate)
• $x$ — the independent quantity (what you control)

If you're earning $15 per hour:

• $x$ = hours worked (you decide this)
• $k$ = 15 (fixed rate)
• $y$ = total pay (depends on $x$)

$y = 15x$

Work 5 hours? y = 15(5) = \75$.

Why There's No $+ b$

This is the most important detail. In $y = kx$, when $x = 0$, $y = 0$:

$y = k(0) = 0$

If the equation had a $+ b$, like $y = kx + b$, then when $x = 0$:

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

A graph not passing through the origin. That extra $b$ means something is already there before you start — which breaks proportionality.

Proportional: You earn $0 when you work 0 hours. Not proportional: You get a $20 signing bonus plus $15 per hour.

Connecting Tables, Graphs, and Equations

The power of $y = kx$ is that it unifies all three representations.

| Representation | Form | Example | |---|---|---| | Equation | $y = kx$ | $y = 4x$ | | Table | Each $y$ is $k$ times $x$ | $(1,4), (2,8), (3,12)$ | | Graph | Straight line through $(0, 0)$ | Slope $= k = 4$ |

If any representation gives you a different $k$, something is wrong.

Finding $k$ in $y = kx$

Rearrange the equation:

$k = \frac{y}{x}$

For any point $(x, y)$ on a proportional relationship (except where $x = 0$), dividing $y$ by $x$ gives $k$.

A table shows $(5, 20)$. $k = 20 \div 5 = 4$. The equation is $y = 4x$.

A graph passes through $(3, 21)$. $k = 21 \div 3 = 7$. The equation is $y = 7x$.

What $k$ Looks Like

• $k$ can be a whole number: $y = 5x$
• $k$ can be a fraction: $y = \frac{3}{4}x$
• $k$ can be a decimal: $y = 0.25x$
• $k$ can be negative: $y = -2x$

Every single one is a proportional relationship.

FAQ

What does $y = kx$ represent in real life? Any situation where one quantity is a fixed multiple of another: pay by the hour, distance at constant speed, cost per item.

Is $y = kx$ the same as $y = mx$? Yes. They're identical. In proportional relationships, $m$ (slope) equals $k$ (constant of proportionality). Different letters, same meaning.

What if the equation is $y = 5x + 2$? Not proportional. The $+2$ shifts the line off the origin. This is a linear relationship but not a proportional one.

Can $k$ be negative in $y = kx$? Yes. A negative $k$ means $y$ decreases when $x$ increases. The line still goes through the origin — it just points downward.

Why does the line always go through $(0, 0)$? Because $k \times 0 = 0$. When you have none of $x$, you get none of $y$.

Related: Use the constant of proportionality calculator to find $k$, or check if an equation is proportional automatically.