Question

for each table, determine whether it shows a direct variation. if it does, write its direct variation equation.

x	y		not direct variation		x	y		not direct variation
-7	8.75		direct variation		3	6		direct variation
-4	5		equation:		6	12		equation:
8	-10				12	36

Explanation:

Step1: Analyze the first table (x, y: -7,8.75; -4,5; 8,-10)

For direct variation, \( y = kx \), so \( k=\frac{y}{x} \) should be constant.

• For \( x=-7, y = 8.75 \): \( k=\frac{8.75}{-7}=-1.25 \)
• For \( x=-4, y = 5 \): \( k=\frac{5}{-4}=-1.25 \)
• For \( x=8, y = -10 \): \( k=\frac{-10}{8}=-1.25 \)

Since \( k \) is constant (\( k=-1.25 \) or \( -\frac{5}{4} \)), it is direct variation. Equation: \( y = -\frac{5}{4}x \) (or \( y=-1.25x \)).

Step2: Analyze the second table (x, y: 3,6; 6,12; 12,36)

Check \( k=\frac{y}{x} \):

• For \( x=3, y = 6 \): \( k=\frac{6}{3}=2 \)
• For \( x=6, y = 12 \): \( k=\frac{12}{6}=2 \)
• For \( x=12, y = 36 \): \( k=\frac{36}{12}=3 \)

Since \( k \) is not constant (2 vs. 3), it is not direct variation.

Answer:

First table: Direct variation, Equation: \( y = -\frac{5}{4}x \) (or \( y = -1.25x \))
Second table: Not direct variation