Question

for each equation decide whether it is proportional or not. then explain why.

1. ( y = 5x )

yes / no
explain _____________

1. ( k = 3t + 7 )

yes / no
explain _____________

1. ( 3w + 5 = z )

yes / no
explain _____________

1. ( w = 7.5b )

yes / no
explain _____________

1. ( 6p = m )

yes / no
explain _____________

1. ( 2g = 4e )

yes / no
explain _____________

for each table decide whether it is proportional or not. then explain why.

1. yes / no

explain _____________
\begin{tabular}{|l|l|l|l|l|}
hline time & 0 & 1 & 2 & 3 & 4 \\
hline distance & 0 & 3 & 6 & 9 & 12 \\
hline
end{tabular}

1. yes / no

explain _____________
\begin{tabular}{|l|l|l|l|l|}
hline months & 0 & 1 & 2 & 3 & 4 \\
hline savings $ & 8 & 10 & 12 & 14 & 16 \\
hline
end{tabular}

1. yes / no

explain _____________
\begin{tabular}{|l|l|l|l|l|}
hline hours & 0 & 1 & 2 & 3 & 4 \\
hline distance & 0 & 15 & 25 & 35 & 45 \\
hline
end{tabular}

1. yes / no

explain _____________
\begin{tabular}{|l|l|l|l|l|}
hline lemons & 4 & 8 & 12 & 16 & 20 \\
hline pies & 1 & 2 & 3 & 4 & 5 \\
hline
end{tabular}

Explanation:

Step1: Recall Proportional Definition

A relationship \( y = kx \) (or equivalent) is proportional if it passes through \((0,0)\) and has constant ratio \( \frac{y}{x}=k \) (or for other variables). For equations, check if it's in \( y = kx \) form (no constant term). For tables, check if \( \frac{\text{output}}{\text{input}} \) is constant and when input is 0, output is 0.

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

• Form: \( y = kx \) ( \( k = 5 \) ), no constant term.
• At \( x = 0 \), \( y = 0 \). Ratio \( \frac{y}{x}=5 \) (constant). So proportional.

Step3: Analyze Equation 9 (\( k = 3i + 7 \))

• Form: \( k = 3i + 7 \), has constant term 7. At \( i = 0 \), \( k = 7

eq 0 \). Not in \( y = kx \) form. So not proportional.

Step4: Analyze Equation 10 (\( 3w + 5 = z \))

• Form: \( z = 3w + 5 \), constant term 5. At \( w = 0 \), \( z = 5

eq 0 \). Not proportional.

Step5: Analyze Equation 11 (\( w = 7.5b \))

• Form: \( w = kb \) ( \( k = 7.5 \) ), no constant term. At \( b = 0 \), \( w = 0 \). Ratio \( \frac{w}{b}=7.5 \) (constant). Proportional.

Step6: Analyze Equation 12 (\( 6p = m \))

• Rewrite: \( m = 6p \), form \( m = kp \) ( \( k = 6 \) ). At \( p = 0 \), \( m = 0 \). Ratio \( \frac{m}{p}=6 \) (constant). Proportional.

Step7: Analyze Equation 13 (\( 2g = 4e \))

• Rewrite: \( g = 2e \) (divide both sides by 2), form \( g = ke \) ( \( k = 2 \) ). At \( e = 0 \), \( g = 0 \). Ratio \( \frac{g}{e}=2 \) (constant). Proportional.

Step8: Analyze Table 14 (Time vs Distance: Time 0 - 4, Distance 0, 3, 6, 9, 12)

• Input 0: Output 0. Ratios: \( \frac{3}{1}=3 \), \( \frac{6}{2}=3 \), \( \frac{9}{3}=3 \), \( \frac{12}{4}=3 \). Constant ratio 3. So proportional.

Step9: Analyze Table 15 (Months vs Savings: Months 0 - 4, Savings 8, 10, 12, 14, 16)

• Input 0: Output 8 ≠ 0. Ratio \( \frac{10}{1}=10 \), \( \frac{12}{2}=6 \) (not constant). Not proportional.

Step10: Analyze Table 16 (Hours vs Distance: Hours 0 - 4, Distance 0, 15, 25, 35, 45)

• Input 0: Output 0. Ratios: \( \frac{15}{1}=15 \), \( \frac{25}{2}=12.5 \) (not constant). Not proportional.

Step11: Analyze Table 17 (Lemons vs Pies: Lemons 4,8,12,16,20; Pies 1,2,3,4,5)

• Input (lemons) 4: output (pies) 1. Ratio \( \frac{1}{4}=0.25 \), \( \frac{2}{8}=0.25 \), \( \frac{3}{12}=0.25 \), etc. Constant ratio 0.25. When lemons = 0 (extrapolate), pies = 0. So proportional.

Answer:

• 8) yes; Explanation: \( y = 5x \) is \( y = kx \) (no constant, passes through (0,0), constant ratio).
• 9) no; Explanation: \( k = 3i + 7 \) has constant term 7, not \( y = kx \) form.
• 10) no; Explanation: \( z = 3w + 5 \) has constant term 5, not \( y = kx \) form.
• 11) yes; Explanation: \( w = 7.5b \) is \( w = kb \) (no constant, constant ratio).
• 12) yes; Explanation: \( m = 6p \) is \( m = kp \) (no constant, constant ratio).
• 13) yes; Explanation: \( g = 2e \) (simplified) is \( g = ke \) (no constant, constant ratio).
• 14) yes; Explanation: Distance/Time = 3 (constant), (0,0) included.
• 15) no; Explanation: Savings at 0 months is 8 (not 0), ratio not constant.
• 16) no; Explanation: Distance/Hours ratios (15, 12.5, ...) not constant.
• 17) yes; Explanation: Pies/Lemons = 0.25 (constant), (0,0) extrapolates.