- Verify the constant ratio \(\frac{y}{x} = -3\) for all given coordinate pairs \((1, -3), (3, -9), (5, -15)\).
- Confirm that the constant of variation is \(k = -3\).
- State the equation of direct variation as \(y = -3x\).
- Select the correct option: \(\text{Yes; } k = -3 \text{ and } y = -3x\).
Explanation
- Analyzing the data and definition of direct variation
To determine whether \(y\) varies directly with \(x\), we check if there exists a constant \(k\) such that \(y = kx\) for all pairs \((x, y)\) in the given table. This is equivalent to checking if the ratio \(\frac{y}{x}\) is constant for every pair.
The given pairs are:
- \(x = 1, y = -3\)
- \(x = 3, y = -9\)
- \(x = 5, y = -15\)
- Calculating the ratio and determining the constant of variation
For each pair \((x, y)\), we calculate the ratio \(k = \frac{y}{x}\):
- For \((1, -3)\): \(k_1 = \frac{-3}{1} = -3\)
- For \((3, -9)\): \(k_2 = \frac{-9}{3} = -3\)
- For \((5, -15)\): \(k_3 = \frac{-15}{5} = -3\)
Since the ratio is the same for all pairs, \(k_1 = k_2 = k_3 = -3\), we conclude that \(y\) varies directly with \(x\). The constant of variation is \(k = -3\).
- Final conclusion and selection of the correct option
Given that \(k = -3\), the equation relating \(x\) and \(y\) is \(y = -3x\). Comparing this result to the provided options, it matches "Yes; \(k = -3\) and \(y = -3x\)". Therefore, the final answer is that \(y\) varies directly with \(x\) with \(k = -3\) and the equation \(y = -3x\).
Examples
Direct variation is a fundamental concept in modeling real-world relationships where one quantity changes in proportion to another. For example, if you are buying items at a constant price, the total cost \(y\) varies directly with the number of items \(x\) you purchase, where the price per item is the constant \(k\). By identifying this proportionality, you can easily predict costs or plan budgets for various quantities. This principle applies to many fields, from physics (like Hooke's Law) to everyday finance and resource management.