Question

determine whether y varies directly with x. if so, find the constant of variation k and write the equation.

Answer 1

yes;k=1;y=x
yes;k=2;y=2x
yes;k=3;y=3x
no

Answer 2

We say that \(y\) varies directly with \(x\) if we can express \(y\) as a constant multiple of \(x\); as an equation, that would mean \(y=kx\) where \(k\) is commonly called the constant of proportionality/variation. We can rewrite this equation as \(\dfrac{y}{x} =k\) (simply by dividing the original equation through by \(x\)).

So, to see if this is a direct variation, you'll want to evaluate \(\dfrac{4}{2}\), \(\dfrac{12}{6}\), \(\dfrac{20}{10}\), and \(\dfrac{28}{14}\). If they all reduce to the same number, then this will be the \(k\) value in the equation \(y=kx\).

Do you think you can take things from here? :-)