Question

for each equation, determine whether x and y are directly proportional (that is, if the equation shows
if so, then find the constant of proportionality (the constant of variation).
(a) (-x = y)
\t● proportional
\t\tconstant of proportionality: (k = square)
\t○ not proportional
(b) (y = \frac{7}{5}x)
\t● proportional
\t\tconstant of proportionality: (k = square)
\t○ not proportional

Explanation:

Part (a)

Step 1: Recall Direct Proportion Formula

Direct proportionality is in the form \( y = kx \), where \( k \) is the constant of proportionality.

Step 2: Analyze the Equation \(-x = y\)

Rewrite the equation as \( y = -1 \cdot x \). This is in the form \( y = kx \) with \( k = -1 \). So \( x \) and \( y \) are directly proportional.

Part (b)

Step 1: Recall Direct Proportion Formula

Direct proportionality is in the form \( y = kx \), where \( k \) is the constant of proportionality.

Step 2: Analyze the Equation \( y=\frac{7}{5}x \)

The equation is already in the form \( y = kx \) where \( k=\frac{7}{5} \). So \( x \) and \( y \) are directly proportional.

Answer:

(a) Constant of proportionality: \( k = -1 \)
(b) Constant of proportionality: \( k=\frac{7}{5} \)