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. 8y = 5x \bigcirc direct variation \quad \quad \quad \quad \quad \quad k = \square \bigcirc not direct variation -2x + 5y = -6 \bigcirc direct variation \quad \quad \quad \quad \quad \quad k = \square \bigcirc not direct variation

Explanation:

Step1: Recall direct variation form

The form of direct variation is \( y = kx \), where \( k \) is the constant of variation. We need to rewrite each equation in this form (or see if it can be) and check if there's a non - zero constant \( k \) and no additional constant term.

Step2: Analyze \( 8y = 5x \)

To get it in the form \( y=kx \), we solve for \( y \). Divide both sides of \( 8y = 5x \) by 8:
\( y=\frac{5}{8}x \)
This is in the form \( y = kx \) with \( k=\frac{5}{8} \), so it is a direct variation.

Step3: Analyze \( - 2x+5y=-6 \)

We solve for \( y \). First, add \( 2x \) to both sides:
\( 5y = 2x-6 \)
Then divide both sides by 5:
\( y=\frac{2}{5}x-\frac{6}{5} \)
Since there is a constant term \( -\frac{6}{5} \) (other than 0), this equation is not in the form \( y = kx \), so it is not a direct variation.

Answer:

For \( 8y = 5x \): Direct variation, \( k=\frac{5}{8} \)
For \( -2x + 5y=-6 \): Not direct variation