Roni wants to write an equation to represent a proportional relationship that has a constant of proportionality equal to [tex]\(\frac{7}{25}\)[/tex]. She writes the equation [tex]\(y = x + \frac{7}{25}\)[/tex].
Let's analyze the problem step-by-step:
1. Understanding Proportional Relationships:
- A proportional relationship between two variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] means that as [tex]\(x\)[/tex] changes, [tex]\(y\)[/tex] changes in such a way that their ratio remains constant. This can be represented as [tex]\(y = kx\)[/tex], where [tex]\(k\)[/tex] is the constant of proportionality.
2. Constant of Proportionality:
- In this case, the constant of proportionality is given as [tex]\(\frac{7}{25}\)[/tex]. Therefore, the correct equation representing this relationship should be [tex]\(y = \frac{7}{25} x\)[/tex].
3. Analyzing Roni's Equation:
- Roni wrote the equation as [tex]\(y = x + \frac{7}{25}\)[/tex]. This equation represents a linear relationship, but it is not a proportional relationship because it includes an additive term [tex]\(\frac{7}{25}\)[/tex] instead of a multiplicative constant.
4. Matching the Options:
- The first incorrect option proposes [tex]\(y = -x + \frac{7}{25}\)[/tex], suggesting a different type of relationship where [tex]\(y\)[/tex] has a constant sum with [tex]\(x\)[/tex]. This is not relevant to a proportional relationship.
- The second incorrect option proposes [tex]\(xy = \frac{7}{25}\)[/tex], indicating a relationship where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] have a constant product. This does not describe the proportional relationship in question.
- The correct option proposes [tex]\(y = \frac{7}{25} x\)[/tex], which matches the definition of a proportional relationship with a constant of proportionality [tex]\(\frac{7}{25}\)[/tex]. This correctly ensures that the ratio [tex]\(y/x\)[/tex] remains constant at [tex]\(\frac{7}{25}\)[/tex].
- The fourth incorrect option suggests [tex]\(y = \frac{7}{25}\)[/tex], which indicates that [tex]\(y\)[/tex] is constant for all values of [tex]\(x\)[/tex]. This does not correctly describe a proportional relationship.
Based on the analysis, the error Roni is making is that she wrote the equation as [tex]\(y = x + \frac{7}{25}\)[/tex] instead of the correct form for a proportional relationship, which should be [tex]\(y = \frac{7}{25} x\)[/tex]. Therefore, the right choice is:
She should have written [tex]\(y = \frac{7}{25} x\)[/tex] so that [tex]\(x\)[/tex] and [tex]\(y\)[/tex] have a constant quotient.
