Respuesta :
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:
[tex]y=mx+b[/tex]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:
[tex]y=mx[/tex]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:
[tex]m=\frac{0-(-4)}{0-(-1)}=\frac{0+4}{0+1}=\frac{4}{1}=4[/tex]We have calculated the slope to be 4, then the equation representing the direct variation is:
[tex]y=4x[/tex]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:
[tex]\begin{gathered} 0=4\cdot8 \\ 0=16 \end{gathered}[/tex]8,0 does not satisfy, therefore it is not an element of the direct variation.
2,8:
[tex]\begin{gathered} 8=4\cdot2 \\ 8=8 \end{gathered}[/tex]2,8 is element of the dierct variation
-2,0:
[tex]\begin{gathered} 0=4\cdot(-2) \\ 0=-8 \end{gathered}[/tex]-2,0 is not part
4,-1:
[tex]\begin{gathered} -1=4\cdot4 \\ -1=16 \end{gathered}[/tex]4,-1 is not part
8,-1:
[tex]\begin{gathered} -1=4\cdot8 \\ -1=32 \end{gathered}[/tex]8,-1 is not part
-2,-8:
[tex]\begin{gathered} -8=4\cdot(-2) \\ -8=-8 \end{gathered}[/tex]-2,-8 is part.
Finally, we can say points (-4,-1), (2,8) and (-2,-8) are part of the direct variation.
