Question

for each equation, determine whether it shows a direct variation (if it does, find the constant of variation and write it in simplest form
-8x + 3y = 7
○ direct variation
k =
○ not direct variation
4x - 5y = 0
direct variation
k =
○ not direct variation

Explanation:

Step1: Analyze \(-8x + 3y = 7\)

A direct variation equation has the form \(y = kx\) (where \(k\) is a constant) and passes through the origin \((0,0)\). Let's rewrite \(-8x + 3y = 7\) in slope - intercept form (\(y=mx + b\)).
Add \(8x\) to both sides: \(3y=8x + 7\).
Divide both sides by 3: \(y=\frac{8}{3}x+\frac{7}{3}\).
Since \(b = \frac{7}{3}
eq0\), this equation does not represent a direct variation.

Step2: Analyze \(4x - 5y = 0\)

Rewrite the equation in slope - intercept form.
Subtract \(4x\) from both sides: \(-5y=-4x\).
Divide both sides by \(- 5\): \(y=\frac{4}{5}x\).
This is in the form \(y = kx\) where \(k=\frac{4}{5}\), so it is a direct variation with \(k = \frac{4}{5}\).

Answer:

For \(-8x + 3y = 7\): Not direct variation
For \(4x - 5y = 0\): Direct variation, \(k=\frac{4}{5}\)