To determine if the relationship between the number of items [tex]\( x \)[/tex] and the amount paid [tex]\( y \)[/tex] forms a direct variation, we need to check if the ratio [tex]\(\frac{y}{x}\)[/tex] is constant for all data points. In mathematical terms, a direct variation relationship can be expressed as [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
Here are the steps to verify this:
1. Calculate the ratio [tex]\(\frac{y}{x}\)[/tex] for each data point:
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 0.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{0.50}{1} = 0.50
\][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 1.00 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{1.00}{2} = 0.50
\][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 1.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{1.50}{3} = 0.50
\][/tex]
- For [tex]\( x = 5 \)[/tex] and [tex]\( y = 2.50 \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{2.50}{5} = 0.50
\][/tex]
2. Examine the calculated ratios to determine if they are all the same:
- The ratios are [tex]\( 0.50, 0.50, 0.50, 0.50 \)[/tex], which are indeed all equal.
3. Conclusion:
Since the ratio [tex]\(\frac{y}{x}\)[/tex] is constant for all data points, [tex]\( y\)[/tex] and [tex]\(x \)[/tex] have a direct variation relationship. The constant of proportionality [tex]\( k \)[/tex] is [tex]\( 0.50 \)[/tex].
Hence, the relationship [tex]\( y = kx \)[/tex] where [tex]\( k = 0.50 \)[/tex] is confirmed for all the given data points.
Therefore, this relationship forms a direct variation and the verification confirms this.
