Math Problem Statement

find the constant of variation k and the undetermined value in the table if y is directly proportional to x.

Solution

To find the constant of variation kk and the undetermined value in a table when yy is directly proportional to xx, we can use the direct variation relationship formula:

y=kxy = kx

Here, kk is the constant of variation, and yy varies directly with xx.

Steps to Solve the Problem:

  1. Identify the Relationship: Since yy is directly proportional to xx, we can write: y=kxy = kx

  2. Find kk: To find kk, we need at least one pair of xx and yy values from the table.

  3. Calculate the Undetermined Value: Once we find kk, we can use it to calculate the missing yy or xx value.

Example Problem

Let’s say we have the following table with some values missing:

\hline x & y \\ \hline 2 & 10 \\ 4 & y \\ x & 30 \\ \hline \end{array}$$ #### 1. Find the constant $$k$$ Using the pair $$(x, y) = (2, 10)$$, we can substitute into the direct variation formula: $$y = kx \Rightarrow 10 = k \times 2$$ $$k = \frac{10}{2} = 5$$ So, the constant of variation $$k$$ is 5. #### 2. Calculate the undetermined value $$y$$ when $$x = 4$$ Using the direct variation formula with $$k = 5$$: $$y = kx \Rightarrow y = 5 \times 4 = 20$$ Thus, $$y = 20$$ when $$x = 4$$. #### 3. Calculate the undetermined value $$x$$ when $$y = 30$$ Again, using the formula $$y = kx$$ with $$k = 5$$: $$30 = 5x$$ $$x = \frac{30}{5} = 6$$ Thus, $$x = 6$$ when $$y = 30$$. ### Conclusion - The constant of variation $$k$$ is 5. - When $$x = 4$$, $$y = 20$$. - When $$y = 30$$, $$x = 6$$. --- Would you like more details on this process, or do you have any specific questions? ### Related Questions: 1. How would the calculation change if $$y$$ were inversely proportional to $$x$$? 2. What are some examples of real-life situations where direct variation is applicable? 3. How do we handle cases where the relationship is not directly proportional but instead involves a more complex equation? 4. What if the constant of variation $$k$$ was a fraction? How would that affect the calculations? 5. How can we graph the relationship between $$x$$ and $$y$$ for a direct variation? **Tip:** Direct variation problems are easier to solve when you identify pairs of known values for $$x$$ and $$y$$ because they allow you to determine the constant $$k$$, which you can then use to find other missing values.

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Math Problem Analysis

Mathematical Concepts

Direct Variation
Proportional Relationships

Formulas

Direct Variation Formula: y = kx

Theorems

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Suitable Grade Level

Grades 7-9