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To determine which relationships have the same constant of proportionality as the equation [tex]\( y = \frac{5}{2} x \)[/tex], we need to analyze each given option.
### Option A: [tex]\( 5y = 2x \)[/tex]
We need to rewrite this equation in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality:
[tex]\[ 5y = 2x \][/tex]
[tex]\[ y = \frac{2}{5} x \][/tex]
Here, the constant of proportionality [tex]\(k\)[/tex] is [tex]\( \frac{2}{5} \)[/tex]. This is not equal to [tex]\( \frac{5}{2} \)[/tex]. So, this option does not have the same constant of proportionality.
### Option B: [tex]\( 8y = 20x \)[/tex]
Rewrite this equation in the form [tex]\( y = kx \)[/tex]:
[tex]\[ 8y = 20x \][/tex]
[tex]\[ y = \frac{20}{8} x \][/tex]
[tex]\[ y = \frac{5}{2} x \][/tex]
Here, the constant of proportionality [tex]\(k\)[/tex] is [tex]\( \frac{5}{2} \)[/tex], which is the same as the equation [tex]\( y = \frac{5}{2} x \)[/tex]. So, this option does have the same constant of proportionality.
### Option C
We are given a table of values:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 1 & 2 \frac{1}{2} \\ 4 & 10 \\ 7 & 17 \frac{1}{2} \\ \hline \end{array} \][/tex]
Let's check these values to see if the relationship [tex]\( y = \frac{5}{2} x \)[/tex] holds:
For [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \frac{1}{2} \)[/tex]:
[tex]\[ y = 2.5 = \frac{5}{2} \cdot 1 \][/tex]
This holds true.
For [tex]\( x = 4 \)[/tex] and [tex]\( y = 10 \)[/tex]:
[tex]\[ y = 10 = \frac{5}{2} \cdot 4 \][/tex]
This holds true.
For [tex]\( x = 7 \)[/tex] and [tex]\( y = 17 \frac{1}{2} \)[/tex]:
[tex]\[ y = 17.5 = \frac{5}{2} \cdot 7 \][/tex]
This holds true.
Since all these calculations confirm that [tex]\( y = \frac{5}{2} x \)[/tex], it shows the same constant of proportionality.
### Conclusion
Therefore, the relationships that have the same constant of proportionality as [tex]\( y = \frac{5}{2} x \)[/tex] are:
- (B) [tex]\( 8y = 20x \)[/tex]
- (C) The table of values
Thus, the correct answers are:
[tex]\[ \text{(B), (C)} \][/tex]
### Option A: [tex]\( 5y = 2x \)[/tex]
We need to rewrite this equation in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of proportionality:
[tex]\[ 5y = 2x \][/tex]
[tex]\[ y = \frac{2}{5} x \][/tex]
Here, the constant of proportionality [tex]\(k\)[/tex] is [tex]\( \frac{2}{5} \)[/tex]. This is not equal to [tex]\( \frac{5}{2} \)[/tex]. So, this option does not have the same constant of proportionality.
### Option B: [tex]\( 8y = 20x \)[/tex]
Rewrite this equation in the form [tex]\( y = kx \)[/tex]:
[tex]\[ 8y = 20x \][/tex]
[tex]\[ y = \frac{20}{8} x \][/tex]
[tex]\[ y = \frac{5}{2} x \][/tex]
Here, the constant of proportionality [tex]\(k\)[/tex] is [tex]\( \frac{5}{2} \)[/tex], which is the same as the equation [tex]\( y = \frac{5}{2} x \)[/tex]. So, this option does have the same constant of proportionality.
### Option C
We are given a table of values:
[tex]\[ \begin{array}{|cc|} \hline x & y \\ \hline 1 & 2 \frac{1}{2} \\ 4 & 10 \\ 7 & 17 \frac{1}{2} \\ \hline \end{array} \][/tex]
Let's check these values to see if the relationship [tex]\( y = \frac{5}{2} x \)[/tex] holds:
For [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \frac{1}{2} \)[/tex]:
[tex]\[ y = 2.5 = \frac{5}{2} \cdot 1 \][/tex]
This holds true.
For [tex]\( x = 4 \)[/tex] and [tex]\( y = 10 \)[/tex]:
[tex]\[ y = 10 = \frac{5}{2} \cdot 4 \][/tex]
This holds true.
For [tex]\( x = 7 \)[/tex] and [tex]\( y = 17 \frac{1}{2} \)[/tex]:
[tex]\[ y = 17.5 = \frac{5}{2} \cdot 7 \][/tex]
This holds true.
Since all these calculations confirm that [tex]\( y = \frac{5}{2} x \)[/tex], it shows the same constant of proportionality.
### Conclusion
Therefore, the relationships that have the same constant of proportionality as [tex]\( y = \frac{5}{2} x \)[/tex] are:
- (B) [tex]\( 8y = 20x \)[/tex]
- (C) The table of values
Thus, the correct answers are:
[tex]\[ \text{(B), (C)} \][/tex]
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