Select all of the equations below in which $v$ is directly proportional to $w$. \[ v=\frac{3}{w} \quad v=w+3 \quad v=\frac{w}{3} \] \[ v=3 w \quad v=w^{3} \]
Solution
Step 1 ：Analyze each equation to determine if it can be written in the form \(v = kw\), where \(k\) is a constant.
Step 2 ：\(v = \frac{3}{w}\) cannot be written in the form \(v = kw\), so it does not represent direct proportionality.
Step 3 ：\(v = w + 3\) cannot be written in the form \(v = kw\), so it does not represent direct proportionality.
Step 4 ：\(v = \frac{w}{3}\) can be written as \(v = \frac{1}{3}w\), which is in the form \(v = kw\). Therefore, it represents direct proportionality.
Step 5 ：\(v = 3w\) is already in the form \(v = kw\), so it represents direct proportionality.
Step 6 ：\(v = w^3\) cannot be written in the form \(v = kw\), so it does not represent direct proportionality.
Step 7 ：\(\boxed{v = \frac{w}{3}, v = 3w}\) are the equations in which \(v\) is directly proportional to \(w\).
