Question

for each table, determine whether it shows a direct variation. if it does, write its direct variation equation. two tables with x and y values and options for direct or not direct variation, and equation boxes

Explanation:

Step1: Recall Direct Variation Definition

Direct variation means \( y = kx \) (or \( \frac{y}{x}=k \), constant). For each table, check \( \frac{y}{x} \) for all pairs.

Step2: Analyze First Table (x: 5,15,25; y:4,12,20)

• For \( x=5, y=4 \): \( \frac{y}{x}=\frac{4}{5}=0.8 \)
• For \( x=15, y=12 \): \( \frac{12}{15}=0.8 \)
• For \( x=25, y=20 \): \( \frac{20}{25}=0.8 \)

Constant \( k = 0.8=\frac{4}{5} \), so direct variation. Equation: \( y=\frac{4}{5}x \)

Step3: Analyze Second Table (x:2,4,6; y:5,10,15)

• For \( x=2, y=5 \): \( \frac{5}{2}=2.5 \)
• For \( x=4, y=10 \): \( \frac{10}{4}=2.5 \)
• For \( x=6, y=15 \): \( \frac{15}{6}=2.5 \)

Constant \( k = 2.5=\frac{5}{2} \), so direct variation. Equation: \( y=\frac{5}{2}x \)

Answer:

First Table:

• Direct variation (selected)
• Equation: \( y = \frac{4}{5}x \)

Second Table:

• Direct variation (selected)
• Equation: \( y = \frac{5}{2}x \)