Equation of direct proportion

If \(y\) is directly proportional to \(x\), we write \(y\propto x\). The ratio \(\dfrac{y}{x}\) is a constant \(k\), called the constant of proportionality, so \(y=kx\), where \(k\ne0\).

Why it helps: rearranging \(\dfrac{y}{x}=k\) gives one formula, \(y=kx\), that works for every pair of values. The constant \(k\) is the amount of \(y\) for each unit of \(x\), so it often has units, such as dollars per kilogram.

Method

1. Write \(y=kx\) using the letters in the question.
2. Substitute the given pair of values and solve for \(k\).
3. Rewrite the equation with the value of \(k\).
4. Substitute the new value of \(x\) or \(y\) and solve.

Example

The mass, \(m\) grams, of a copper wire is directly proportional to its length, \(l\) metres. A \(4\) m wire has a mass of \(92\) g. Find an equation connecting \(m\) and \(l\), the mass of \(9\) m of the wire and the length of wire with a mass of \(299\) g.

\(\begin{aligned} m&=kl \\ 92&=k(4) \\ k&=23 \\ \therefore m&=23l \end{aligned}\)

When \(l=9\): \(m=23(9)=207\) g.

When \(m=299\): \(299=23l\), so \(l=\dfrac{299}{23}=13\) m.

Check: substitute the given pair back: \(23\times4=92\). The constant \(k=23\) means \(23\) g for each metre. A common error is to write \(m=\dfrac{k}{l}\), which describes inverse proportion, or to put the wrong letter on the left.