Question

for each equation, determine whether it shows a direct variation (that is, shows directly proportional variables
if it does, find the constant of variation and write it in simplest form.
8x = -7y + 1
○ direct variation
k =
○ not direct variation
16y = 2x
○ direct variation
k =
○ not direct variation

Explanation:

Step1: Recall direct variation form

The equation for direct variation is \( y = kx \), where \( k \) is the constant of variation, and the equation can be written in the form with no constant term (other than 0).

Step2: Analyze \( 8x=-7y + 1 \)

Rearrange the equation \( 8x=-7y + 1 \) to solve for \( y \):

$$
LATEXBLOCK0
$$

Since there is a constant term \( \frac{1}{7} \), it does not fit the form \( y = kx \). So, \( 8x=-7y + 1 \) is not direct variation.

Step3: Analyze \( 16y = 2x \)

Rearrange the equation \( 16y = 2x \) to solve for \( y \):

$$ y=\frac{2}{16}x=\frac{1}{8}x $$

This fits the form \( y = kx \) with \( k=\frac{1}{8} \), so it is direct variation with \( k = \frac{1}{8} \).

Answer:

For \( 8x=-7y + 1 \): Not direct variation
For \( 16y = 2x \): Direct variation, \( k=\frac{1}{8} \)