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Chapter 1: Problem 38

Determine whether the variation model represented by the ordered pairs \((x, y)\) is of the form \(y=k x\) or \(y=k x,\) and find \(k\) Then write a model that relates \(y\) and \(x .\) $$(5,2),(10,4),(15,6),(20,8),(25,10)$$

Short Answer

Expert verified
The variation model is of the form \(y = k \cdot x\) with a constant k = 0.4. The model relating y and x is \(y = 0.4x\).

Step by step solution

01

Determine the Variation Type

Look at the relationship between the x-values and y-values in the ordered pairs. Notice that for every ordered pair, as x increases, so does y, and it seems to be doing so proportionately. As x is not a divisor of y in any of the pairs, but a multiplier, this suggests a directly proportional variation model, or \(y = k \cdot x\).
02

Calculate the Constant of Variation

Substitute one of the ordered pairs into the equation \(y = k \cdot x\). Let's use the first pair \(x = 5, y = 2\). Doing so gives \(2 = k \cdot 5\). Solving for \(k\), divide both sides by 5. This gives \(k = 2 / 5 = 0.4\). To confirm if this value of \(k\) holds for other pairs, substitute remaining ordered pairs into the same equation, replacing \(k\) with 0.4. If all pairs satisfy the equation, it confirms the value of \(k\). Doing this for all remaining pairs, you will see that they all satisfy the equation meaning that \(k = 0.4\) is indeed correct.
03

Write the Model

Using the calculated constant of variation, \(k = 0.4\), and the direct variation form \(y = k \cdot x\), write the model. Plug in the value of \(k\). This gives \(y = 0.4x\). Thus, the model that relates y and x is \(y = 0.4x\).

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