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Sagot :
To determine which of the given relationships have the same constant of proportionality as the equation [tex]\( y = \frac{5}{2} x \)[/tex], we need to compute and compare the constants of proportionality for each option.
### Step-by-Step Solution:
1. Compute the Constant of Proportionality for [tex]\( y = \frac{5}{2} x \)[/tex] (the given relationship):
- The constant of proportionality here is clearly [tex]\(\frac{5}{2} = 2.5\)[/tex].
2. Analyze Option (A): [tex]\( 5y = 2x \)[/tex]
- Rewrite the equation in the form [tex]\( y = kx \)[/tex] to find the constant of proportionality [tex]\( k \)[/tex].
- [tex]\( 5y = 2x \)[/tex]
- Divide both sides by 5 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2}{5} x \][/tex]
- The constant of proportionality is [tex]\(\frac{2}{5} = 0.4\)[/tex].
3. Analyze Option (7): [tex]\( 8y = 20x \)[/tex]
- Rewrite the equation in the form [tex]\( y = kx \)[/tex] to find the constant of proportionality [tex]\( k \)[/tex].
- [tex]\( 8y = 20x \)[/tex]
- Divide both sides by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{20}{8} x \][/tex]
- Simplify [tex]\(\frac{20}{8}\)[/tex]:
[tex]\[ \frac{20}{8} = 2.5 \][/tex]
- The constant of proportionality is 2.5.
4. Analyze the Tabular Data:
- Given pairs [tex]\((x, y)\)[/tex] from the table:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ 1 & 2.5 \\ 4 & 10 \\ 7 & 17.5 \\ \hline \end{array} \][/tex]
- Compute the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For [tex]\( x = 1 \)[/tex]: [tex]\( \frac{2.5}{1} = 2.5 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( \frac{10}{4} = 2.5 \)[/tex]
- For [tex]\( x = 7 \)[/tex]: [tex]\( \frac{17.5}{7} = 2.5 \)[/tex]
- All computed ratios are 2.5, so the tabular data has a constant of proportionality of 2.5.
### Summary of Results:
- Option [tex]\( (A) \)[/tex]: constant of proportionality = 0.4
- Option [tex]\( (7) \)[/tex]: constant of proportionality = 2.5
- Table Data: constant of proportionality = 2.5
Therefore, the relationships that have the same constant of proportionality (2.5) as [tex]\( y = \frac{5}{2} x \)[/tex] are:
1. Option (7) [tex]\( 8y = 20x \)[/tex]
2. The tabular data
[tex]\[ \boxed{(7) \text{ and the tabular data}} \][/tex]
### Step-by-Step Solution:
1. Compute the Constant of Proportionality for [tex]\( y = \frac{5}{2} x \)[/tex] (the given relationship):
- The constant of proportionality here is clearly [tex]\(\frac{5}{2} = 2.5\)[/tex].
2. Analyze Option (A): [tex]\( 5y = 2x \)[/tex]
- Rewrite the equation in the form [tex]\( y = kx \)[/tex] to find the constant of proportionality [tex]\( k \)[/tex].
- [tex]\( 5y = 2x \)[/tex]
- Divide both sides by 5 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2}{5} x \][/tex]
- The constant of proportionality is [tex]\(\frac{2}{5} = 0.4\)[/tex].
3. Analyze Option (7): [tex]\( 8y = 20x \)[/tex]
- Rewrite the equation in the form [tex]\( y = kx \)[/tex] to find the constant of proportionality [tex]\( k \)[/tex].
- [tex]\( 8y = 20x \)[/tex]
- Divide both sides by 8 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{20}{8} x \][/tex]
- Simplify [tex]\(\frac{20}{8}\)[/tex]:
[tex]\[ \frac{20}{8} = 2.5 \][/tex]
- The constant of proportionality is 2.5.
4. Analyze the Tabular Data:
- Given pairs [tex]\((x, y)\)[/tex] from the table:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ 1 & 2.5 \\ 4 & 10 \\ 7 & 17.5 \\ \hline \end{array} \][/tex]
- Compute the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair:
- For [tex]\( x = 1 \)[/tex]: [tex]\( \frac{2.5}{1} = 2.5 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( \frac{10}{4} = 2.5 \)[/tex]
- For [tex]\( x = 7 \)[/tex]: [tex]\( \frac{17.5}{7} = 2.5 \)[/tex]
- All computed ratios are 2.5, so the tabular data has a constant of proportionality of 2.5.
### Summary of Results:
- Option [tex]\( (A) \)[/tex]: constant of proportionality = 0.4
- Option [tex]\( (7) \)[/tex]: constant of proportionality = 2.5
- Table Data: constant of proportionality = 2.5
Therefore, the relationships that have the same constant of proportionality (2.5) as [tex]\( y = \frac{5}{2} x \)[/tex] are:
1. Option (7) [tex]\( 8y = 20x \)[/tex]
2. The tabular data
[tex]\[ \boxed{(7) \text{ and the tabular data}} \][/tex]
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