To determine if \( y \) varies directly with \( x \), we look for a constant ratio \( \frac{y}{x} \) across all given values (except where \( x = 0 \)).
Let's calculate the ratios for the given values of \( x \) and \( y \):
-
For \( x = 1, y = 4 \):
\( \frac{y}{x} = \frac{4}{1} = 4 \) -
For \( x = 2, y = 8 \):
\( \frac{y}{x} = \frac{8}{2} = 4 \) -
For \( x = 3, y = 12 \):
\( \frac{y}{x} = \frac{12}{3} = 4 \)
Since the ratio \( \frac{y}{x} \) is constant and equal to 4 for all values of \( x \) (other than 0), we can conclude that \( y \) does indeed vary directly with \( x \).
The equation for direct variation is \( y = kx \), where \( k \) is the constant of variation. Here, \( k = 4 \).
Thus, the direct variation equation is:
\[ y = 4x \]
So the answer is yes; \( y = 4x \).