Question

I am going to have to send you a photo of the problem during the session because it is to large to crop here.

Answer 1

Direct variations have an special characteristic: they can be represented on a plane by a line paassing through the origin (0,0).

The equation of a line has the following shape:

`y=mx+b`

Where x is the slope, and b is the y intercept.

For direct variations, the line passes through the origin; then, the y intercept is 0, therefore b=0.

For direct variations, we can have an associated line with the following shape:

`y=mx`

We can find the value for m knowing 2 points of the line and calculating the slope. One point is (-1,-4); and the other is the origin (0,0).

Now we can calculate the slope by dividing y distance of the points by the x distance of the points:

`m=(0-(-4))/(0-(-1))=(0+4)/(0+1)=(4)/(1)=4`

We have calculated the slope to be 4, then the equation representing the direct variation is:

`y=4x`

Any pair of points x,y that satisfy the equation will an element of the direct variation.

Now, we can try each:

With 8,0:

`\begin{gathered} 0=4\cdot8 \\ 0=16 \end{gathered}`

8,0 does not satisfy, therefore it is not an element of the direct variation.

2,8:

`\begin{gathered} 8=4\cdot2 \\ 8=8 \end{gathered}`

2,8 is element of the dierct variation

-2,0:

`\begin{gathered} 0=4\cdot(-2) \\ 0=-8 \end{gathered}`

-2,0 is not part

4,-1:

`\begin{gathered} -1=4\cdot4 \\ -1=16 \end{gathered}`

4,-1 is not part

8,-1:

`\begin{gathered} -1=4\cdot8 \\ -1=32 \end{gathered}`

8,-1 is not part

-2,-8:

`\begin{gathered} -8=4\cdot(-2) \\ -8=-8 \end{gathered}`

-2,-8 is part.

Finally, we can say points (-4,-1), (2,8) and (-2,-8) are part of the direct variation.