Question

8.
For the data in the table, tell whether y varies directly with x . If it does, write an equation for the direct variation.


x y
0 0
1 4
2 8
3 12

Answer 1

To determine whether \( y \) varies directly with \( x \), we need to check if there is a constant ratio \( \frac{y}{x} \) for all the values of \( x \) and \( y \).

From the data in the table:

- When \( x = 0 \), \( y = 0 \): \( \frac{y}{x} \) is undefined (but because both are 0, it is consistent with direct variation).
- When \( x = 1 \), \( y = 4 \): \( \frac{y}{x} = \frac{4}{1} = 4 \)
- When \( x = 2 \), \( y = 8 \): \( \frac{y}{x} = \frac{8}{2} = 4 \)
- When \( x = 3 \), \( y = 12 \): \( \frac{y}{x} = \frac{12}{3} = 4 \)

Now, since the ratio \( \frac{y}{x} = 4 \) is constant (for \( x \neq 0 \)), we can conclude that \( y \) varies directly with \( x \).

The equation for the direct variation can be written as:

\[
y = kx
\]

where \( k \) is the constant of variation. Here, \( k = 4 \).

Thus, the equation is:

\[
y = 4x
\]