Question

determine whether y varies directly with x. if so find the constant variation and write the equation x -6, -9, -15; y 2, 3, 5

Answer 1

Answer:

Direct variation; k = -3; y = -3x  

Step-by-step explanation:

The formula for direct variation is

y = kx     Divide both sides by x

k = y/x

For x = -6, y = 2:

k = -6/2

k = -3

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For x = -9, y = 3:

k = -9/3

k = -3

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For x = -15, y = 5:

k = -15/5

k = -3

y = -3x

y and x are always in the same ratio, so y varies directly with x.

Answer 2

$\bf \qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf \begin{array}{|cc|ll} \cline{1-2} x&y\\ \cline{1-2} \stackrel{-3\cdot 2}{-6}&2\\ \stackrel{-3\cdot 3}{-9}&3\\ \stackrel{-3\cdot 5}{-15}&5\\ \cline{1-2} \end{array}~\hspace{5em}\textit{we know that } \begin{cases} y=3\\ x=-9 \end{cases}\implies 3=k(-9) \\\\\\ \cfrac{3}{-9}=k\implies -\cfrac{1}{3}=k\qquad therefore\qquad \blacktriangleright y=-\cfrac{1}{3}x \blacktriangleleft$

as you can see above, we can see that "x" can be a product of whatever "y" happens to be, so we know each variable is just a factor away from the other.