Math Problem Statement

SOLVE

Solution

To determine which representation shows a proportional relationship between xx and yy, we need to identify where yy is directly proportional to xx. This means that for each pair (x,y)(x, y), the ratio yx\frac{y}{x} should be constant.

Let's analyze each option:

Option A: Table

x & y \\ \hline 2 & 1 \\ 4 & 3 \\ 17 & 16 \\ 29 & 28 \\ \end{array}$$ For a proportional relationship, $$\frac{y}{x}$$ should be constant: 1. $$\frac{y}{x} = \frac{1}{2} = 0.5$$ 2. $$\frac{y}{x} = \frac{3}{4} = 0.75$$ 3. $$\frac{y}{x} = \frac{16}{17} \approx 0.941$$ 4. $$\frac{y}{x} = \frac{28}{29} \approx 0.966$$ The ratios are not the same, so **Option A does not represent a proportional relationship**. ### Option B: Table $$\begin{array}{c|c} x & y \\ \hline 9 & 28 \\ 12 & 34 \\ 16 & 52 \\ 28 & 82 \\ \end{array}$$ Let's calculate $$\frac{y}{x}$$ for each pair: 1. $$\frac{y}{x} = \frac{28}{9} \approx 3.11$$ 2. $$\frac{y}{x} = \frac{34}{12} \approx 2.83$$ 3. $$\frac{y}{x} = \frac{52}{16} = 3.25$$ 4. $$\frac{y}{x} = \frac{82}{28} \approx 2.93$$ The ratios are not the same, so **Option B does not represent a proportional relationship**. ### Option C: Graph The graph shows a straight line passing through the origin (0,0) with a positive slope. - A straight line through the origin indicates a proportional relationship where $$y = kx$$ for some constant $$k$$. Thus, **Option C represents a proportional relationship**. ### Option D: Graph The graph shows a straight line, but it does **not** pass through the origin (0,0). - Since it does not go through the origin, **Option D does not represent a proportional relationship**. ### Conclusion: The correct answer is **Option C**, as it represents a proportional relationship between $$x$$ and $$y$$. Would you like more details on any part of this solution or have any questions? Here are 5 related questions you might consider: 1. What is the definition of a proportional relationship in mathematics? 2. How do you determine if a graph represents a proportional relationship? 3. What are some examples of real-world proportional relationships? 4. How can you convert a non-proportional relationship into a proportional one? 5. What is the significance of the slope in a proportional relationship graph? **Tip:** When checking for proportional relationships in a table, always look for a constant ratio $$\frac{y}{x}$$.

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Math Problem Analysis

Mathematical Concepts

Proportional Relationships
Graph Analysis

Formulas

-

Theorems

-

Suitable Grade Level

Grades 6-8

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