Answer :
To determine which relationships have the same constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] as the equation [tex]\( y = \frac{1}{2} x \)[/tex], we need to assess each given option.
1. Option (A): [tex]\( 6y = 3x \)[/tex]
- Rearrange the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 6y = 3x \implies y = \frac{3x}{6} \implies y = \frac{1}{2} x \][/tex]
- The constant of proportionality here is [tex]\( \frac{1}{2} \)[/tex]. This matches the original equation [tex]\( y = \frac{1}{2} x \)[/tex].
2. Option (B)
- Unfortunately, no specific information is provided, so we cannot determine the constant of proportionality for this option.
3. Option (C)
- Similarly, no specific information is provided, so we cannot determine the constant of proportionality for this option.
4. Option (D): Table
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 2 & 1 \\ 3 & 2.5 \\ 4 & 3 \\ \hline \end{array} \][/tex]
- Calculate the constant of proportionality ([tex]\( k \)[/tex]) for each pair of [tex]\( (x, y) \)[/tex]:
[tex]\[ k = \frac{y}{x} \][/tex]
For [tex]\( (2, 1) \)[/tex]:
[tex]\[ k = \frac{1}{2} = \frac{1}{2} \][/tex]
For [tex]\( (3, 2.5) \)[/tex]:
[tex]\[ k = \frac{2.5}{3} \approx 0.833 \neq \frac{1}{2} \][/tex]
For [tex]\( (4, 3) \)[/tex]:
[tex]\[ k = \frac{3}{4} = 0.75 \neq \frac{1}{2} \][/tex]
- Not all rows have the same [tex]\( k \)[/tex]; therefore, this option does not have a consistent constant of proportionality.
5. Option (E): Table
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 4 & 2 \\ 5 & 2.5 \\ 10 & 5 \\ \hline \end{array} \][/tex]
- Calculate the constant of proportionality ([tex]\( k \)[/tex]) for each pair of [tex]\( (x, y) \)[/tex]:
[tex]\[ k = \frac{y}{x} \][/tex]
For [tex]\( (4, 2) \)[/tex]:
[tex]\[ k = \frac{2}{4} = \frac{1}{2} \][/tex]
For [tex]\( (5, 2.5) \)[/tex]:
[tex]\[ k = \frac{2.5}{5} = \frac{1}{2} \][/tex]
For [tex]\( (10, 5) \)[/tex]:
[tex]\[ k = \frac{5}{10} = \frac{1}{2} \][/tex]
- All rows have the same [tex]\( k \)[/tex]; therefore, this option has a consistent constant of proportionality [tex]\( \frac{1}{2} \)[/tex].
Based on this analysis, the choices that have the same constant of proportionality as [tex]\( y = \frac{1}{2} x \)[/tex] are:
- (A) [tex]\( 6y = 3x \)[/tex]
- (E) the table with [tex]\( x = 4, y = 2 \)[/tex], [tex]\( x = 5, y = 2.5 \)[/tex], and [tex]\( x = 10, y = 5 \)[/tex]
Thus, the answers are: [tex]\( \text{(A)} \)[/tex] and [tex]\( \text{(E)} \)[/tex].
1. Option (A): [tex]\( 6y = 3x \)[/tex]
- Rearrange the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 6y = 3x \implies y = \frac{3x}{6} \implies y = \frac{1}{2} x \][/tex]
- The constant of proportionality here is [tex]\( \frac{1}{2} \)[/tex]. This matches the original equation [tex]\( y = \frac{1}{2} x \)[/tex].
2. Option (B)
- Unfortunately, no specific information is provided, so we cannot determine the constant of proportionality for this option.
3. Option (C)
- Similarly, no specific information is provided, so we cannot determine the constant of proportionality for this option.
4. Option (D): Table
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 2 & 1 \\ 3 & 2.5 \\ 4 & 3 \\ \hline \end{array} \][/tex]
- Calculate the constant of proportionality ([tex]\( k \)[/tex]) for each pair of [tex]\( (x, y) \)[/tex]:
[tex]\[ k = \frac{y}{x} \][/tex]
For [tex]\( (2, 1) \)[/tex]:
[tex]\[ k = \frac{1}{2} = \frac{1}{2} \][/tex]
For [tex]\( (3, 2.5) \)[/tex]:
[tex]\[ k = \frac{2.5}{3} \approx 0.833 \neq \frac{1}{2} \][/tex]
For [tex]\( (4, 3) \)[/tex]:
[tex]\[ k = \frac{3}{4} = 0.75 \neq \frac{1}{2} \][/tex]
- Not all rows have the same [tex]\( k \)[/tex]; therefore, this option does not have a consistent constant of proportionality.
5. Option (E): Table
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 4 & 2 \\ 5 & 2.5 \\ 10 & 5 \\ \hline \end{array} \][/tex]
- Calculate the constant of proportionality ([tex]\( k \)[/tex]) for each pair of [tex]\( (x, y) \)[/tex]:
[tex]\[ k = \frac{y}{x} \][/tex]
For [tex]\( (4, 2) \)[/tex]:
[tex]\[ k = \frac{2}{4} = \frac{1}{2} \][/tex]
For [tex]\( (5, 2.5) \)[/tex]:
[tex]\[ k = \frac{2.5}{5} = \frac{1}{2} \][/tex]
For [tex]\( (10, 5) \)[/tex]:
[tex]\[ k = \frac{5}{10} = \frac{1}{2} \][/tex]
- All rows have the same [tex]\( k \)[/tex]; therefore, this option has a consistent constant of proportionality [tex]\( \frac{1}{2} \)[/tex].
Based on this analysis, the choices that have the same constant of proportionality as [tex]\( y = \frac{1}{2} x \)[/tex] are:
- (A) [tex]\( 6y = 3x \)[/tex]
- (E) the table with [tex]\( x = 4, y = 2 \)[/tex], [tex]\( x = 5, y = 2.5 \)[/tex], and [tex]\( x = 10, y = 5 \)[/tex]
Thus, the answers are: [tex]\( \text{(A)} \)[/tex] and [tex]\( \text{(E)} \)[/tex].
