Once you know the constant of proportionality, you can write the whole relationship as one short equation. It works for every row of the table at once.
Key idea
A proportional relationship can be written $y = kx$, where $k = \frac{y}{x}$. Find $k$ from any pair, then write the equation.
Why it works
"Proportional" means each $y$ is $k$ times its $x$. The equation $y = kx$ says exactly that. It also lets you find a $y$ for any $x$, even one that is not in the table.
Worked examples
Worked example: From a table
$y$ is proportional to $x$. Write an equation. $$\begin{array}{l|ccc} x & 2 & 5 & 8 \\ \hline y & 7 & 17.5 & 28 \end{array}$$
- $k = \frac{y}{x} = \frac{7}{2} = 3.5$.
- Check: $3.5 \times 8 = 28$ ✓.
- The equation is $y = 3.5x$.
Answer: $y = 3.5x$
Worked example: From a story
4 movie tickets cost \$34. Let $c$ be the cost in dollars and $n$ the number of tickets. Write an equation for $c$ in terms of $n$.
- Cost per ticket: $k = 34 \div 4 = 8.5$.
- Each ticket adds \$8.50, so $c = 8.5n$.
- Check: $8.5 \times 4 = 34$ ✓.
Answer: $c = 8.5n$
Worked example: Watch the order
$x$: $6, 9, 15$ and $y$: $2, 3, 5$. Write $y = kx$.
- $k = \frac{y}{x} = \frac{2}{6} = \frac{1}{3}$. Here $y$ is smaller than $x$, so $k$ is less than 1.
- The equation is $y = \frac{1}{3}x$. Check: $\frac{1}{3} \times 15 = 5$ ✓.
Answer: $y = \frac{1}{3}x$
Common mistake
Dividing $x$ by $y$ and writing the reciprocal, like $y = 3x$ in the last example. Always check your equation with a row of the table: $3 \times 6 = 18$, not $2$, so $y = 3x$ is wrong.