At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine if the relationship between [tex]\(x\)[/tex] (the number of items) and [tex]\(y\)[/tex] (the amount paid) forms a direct variation, we need to check whether [tex]\(y\)[/tex] is always proportional to [tex]\(x\)[/tex]. In mathematical terms, this means that [tex]\(y = kx\)[/tex] for some constant [tex]\(k\)[/tex], where [tex]\(k\)[/tex] is the same for all data points.
Step-by-Step Solution:
1. Identify the Data Points:
The table provides the following data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0.50 \\ \hline 2 & 1.00 \\ \hline 3 & 1.50 \\ \hline 5 & 2.50 \\ \hline \end{array} \][/tex]
2. Calculate the Ratio [tex]\( \frac{y}{x} \)[/tex] for Each Data Point:
- For [tex]\( x = 1, y = 0.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{0.50}{1} = 0.50 \][/tex]
- For [tex]\( x = 2, y = 1.00 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1.00}{2} = 0.50 \][/tex]
- For [tex]\( x = 3, y = 1.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1.50}{3} = 0.50 \][/tex]
- For [tex]\( x = 5, y = 2.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{2.50}{5} = 0.50 \][/tex]
3. Check the Consistency of Ratios:
Observe that the ratio [tex]\( \frac{y}{x} \)[/tex] is consistent across all data points:
[tex]\[ \frac{y}{x} = 0.50 \][/tex]
Therefore, the ratio remains constant at 0.50.
4. Conclude the Direct Variation:
Since the ratio [tex]\( \frac{y}{x} \)[/tex] is always 0.50 for all given data points, it means [tex]\( y = 0.50x \)[/tex]. This confirms that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] with a constant of proportionality [tex]\( k = 0.50 \)[/tex].
Verification:
Indeed, by verifying through consistent ratio [tex]\( \frac{y}{x} = 0.50 \)[/tex] for all pairs, we conclude that the relationship between the number of items and the amount paid forms a direct variation.
Hence, the relationship forms a direct variation. The answer is [tex]\(\boxed{\text{True}}\)[/tex].
Step-by-Step Solution:
1. Identify the Data Points:
The table provides the following data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0.50 \\ \hline 2 & 1.00 \\ \hline 3 & 1.50 \\ \hline 5 & 2.50 \\ \hline \end{array} \][/tex]
2. Calculate the Ratio [tex]\( \frac{y}{x} \)[/tex] for Each Data Point:
- For [tex]\( x = 1, y = 0.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{0.50}{1} = 0.50 \][/tex]
- For [tex]\( x = 2, y = 1.00 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1.00}{2} = 0.50 \][/tex]
- For [tex]\( x = 3, y = 1.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{1.50}{3} = 0.50 \][/tex]
- For [tex]\( x = 5, y = 2.50 \)[/tex]:
[tex]\[ \frac{y}{x} = \frac{2.50}{5} = 0.50 \][/tex]
3. Check the Consistency of Ratios:
Observe that the ratio [tex]\( \frac{y}{x} \)[/tex] is consistent across all data points:
[tex]\[ \frac{y}{x} = 0.50 \][/tex]
Therefore, the ratio remains constant at 0.50.
4. Conclude the Direct Variation:
Since the ratio [tex]\( \frac{y}{x} \)[/tex] is always 0.50 for all given data points, it means [tex]\( y = 0.50x \)[/tex]. This confirms that [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] with a constant of proportionality [tex]\( k = 0.50 \)[/tex].
Verification:
Indeed, by verifying through consistent ratio [tex]\( \frac{y}{x} = 0.50 \)[/tex] for all pairs, we conclude that the relationship between the number of items and the amount paid forms a direct variation.
Hence, the relationship forms a direct variation. The answer is [tex]\(\boxed{\text{True}}\)[/tex].
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
