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Chapter 11: Problem 38

Suppose the variable \(x\) is proportional to \(1 / y\). What does this tell you about how the numeric value of \(x\) changes as the numeric value of \(y\) changes?

Short Answer

Expert verified
In conclusion, when \(x\) is proportional to \(\frac{1}{y}\), it means that \(x\) and \(y\) are inversely related. As the numeric value of \(y\) increases, the numeric value of \(x\) decreases, and vice versa. Their relationship can be represented using the equation: \(x = k\left(\frac{1}{y}\right)\), where \(k\) is the constant of proportionality.

Step by step solution

01

Understand Proportionality

Proportionality means that one variable is directly related to another by a constant factor. In our case, \(x\) is proportional to \(\frac{1}{y}\). We can represent this relationship using the following equation: \[x = k\left(\frac{1}{y}\right)\] where \(k\) is the constant of proportionality.
02

Examine the Relationship Between \(x\) and \(y\)

Now let's analyze how the numeric value of \(x\) changes as the numeric value of \(y\) changes. Since \(x\) is proportional to \(\frac{1}{y}\), if \(y\) increases, \(\frac{1}{y}\) decreases and if \(y\) decreases, \(\frac{1}{y}\) increases. This relationship can be verified by considering the equation: \[x = k\left(\frac{1}{y}\right)\] When the value of \(y\) increases, the value of \(\frac{1}{y}\) decreases, leading to a decrease in the value of \(x\). Conversely, when the value of \(y\) decreases, the value of \(\frac{1}{y}\) increases, leading to an increase in the value of \(x\).
03

Conclusion

In conclusion, the numeric value of \(x\) is inversely proportional to the numeric value of \(y\). As \(y\) increases, \(x\) decreases, and as \(y\) decreases, \(x\) increases. This relationship can be represented by the equation: \[x = k\left(\frac{1}{y}\right)\] where \(k\) is the constant of proportionality.

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