Question

Which line having the given slope and passing through the given point represents a direct variation?

Answer 1

`m=2/3`,
`(9,6)`

Explanation:

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form
`y/x=k` or
`y=kx`

In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin

Verify each case

case A) For
`m=2/3` point
`(9,6)`

The linear direct variation equation is equal to

`y=(2/3)x`

For
`x=9, y=6`

substitute the value of x and the value of y in the equation and then compare the result

`6=(2/3)(9)`

`6=6` ------> is true

therefore

This line represent a direct variation

case B) For
`m=2/3` point
`(6,9)`

The linear direct variation equation is equal to

`y=(2/3)x`

For
`x=6, y=9`

substitute the value of x and the value of y in the equation and then compare the result

`9=(2/3)(6)`

`9=4` ------> is not true

therefore

This line not represent a direct variation

case C) For
`m=2/3` point
`(9,-6)`

The linear direct variation equation is equal to

`y=(2/3)x`

For
`x=9, y=-6`

substitute the value of x and the value of y in the equation and then compare the result

`-6=(2/3)(9)`

`-6=6` ------> is not true

therefore

This line not represent a direct variation

case D) For
`m=-2/3` point
`(9,6)`

The linear direct variation equation is equal to

`y=-(2/3)x`

For
`x=9, y=6`

substitute the value of x and the value of y in the equation and then compare the result

`6=-(2/3)(9)`

`6=-6` ------> is not true

therefore

This line not represent a direct variation