Question

If y varies directly with x, and y = 45 when x = 5, how do you write the direct linear variation equation and what is the constant?

Answer 1

The constant is 9 so `y=kx ->y=9x`

Explanation:

Basically we are back to the good fashioned ratio problem.

Solving by 'not' using the short cut method. I have opted to do this to demonstrate what is really happening with the numbers.

Let `k` be a constant

Then `y/x=k/1`

So for `y/x->45/5 = k/1`

To find the value of `k` we need to find a way of changing the denominator of 5 into 1 without changing the intrinsic value of `45/5`

Maintaining the proportionality of this relation ship what I am about to do to the top (numerator) I will also do to the bottom (denominator)

Divide top and bottom by 5

`(45-:5)/(5-:5) = k/1" "` This is where the shortcut comes from.

`9/1=k/1 =>k=9`
'~~~~~~~~~~~~~~~~~~~~~~~~~~
So the relation ship is

`y/x=45/5=9/1`

`y/x=9/1`

Multiply both sides by `x`

`yxx x/x=9x`

`yxx1=9x =kx`