9. Direct Variation and Constant of Proportionality

A direct variation relationship has the form:
\[ y=kx. \]
The number \(k\) is the constant of proportionality. It is also the unit rate:
\[ k=\frac{y}{x}. \]

Example: A car travels 60 miles each hour. Distance \(d\) varies directly with time \(t\):
\[ d=60t. \]
Here, \(k=60\) miles per hour.

A table is proportional if \(y/x\) is the same for every row. For:
\[ (2,10),\ (4,20),\ (6,30), \]
the value \(y/x\) is 5 each time, so \(y=5x\).

If a graph or table does not pass through the origin or does not keep the same \(y/x\), it is not a direct variation.

Common misconception: thinking any straight line is proportional. A proportional line must go through \((0,0)\).

Academic habit: compute \(y/x\) for at least two rows before deciding a table is proportional.