Finding a Formula for Direct Variation

Property If we know any one pair of corresponding values for the variables in a direct variation, we can find the constant of variation, $k$. To do this, substitute the known values of $x$ and $y$ into the equation $y = kx$ and solve for $k$. Once $k$ is found, you can write the complete formula.

Examples The speed of a falling object, $v$, varies directly with time, $t$. If its speed is 49 meters per second after 5 seconds, we find $k$ from $49 = k(5)$, so $k = 9.8$. The formula is $v = 9.8t$. The property tax, $T$, on a home varies directly with its assessed value, $V$. A home valued at 200,000 dollars has a tax of 3,000 dollars. We have $3000 = k(200000)$, so $k = 0.015$. The formula is $T = 0.015V$. The weight of a bag of apples, $w$, is proportional to the number of apples, $n$. A bag with 10 apples weighs 3 pounds. We find $k$ from $3 = k(10)$, so $k = 0.3$. The formula is $w = 0.3n$.

Explanation To find the specific 'rule' connecting two variables, you only need one matched pair of data. By plugging this pair into the general formula $y = kx$ or $y=kx^n$, you can solve for the constant, $k$, and unlock the full equation.