Chapter 8.5, Problem 16HP

To determine

whether the following statement is always, sometimes, or never true.

Answer to Problem 16HP

The conclusion is that sometimes a linear relationship that has a constant rate of change also shows proportional relationship.

Explanation of Solution

Given: A linear relationship that has a constant rate of change is a proportional relationship.

Calculation:

Consider an example that shows three points: $(10,15),(20,20)$ and $(30,25)$

In order to find constant rate of change, take any two points and use the formula for rate of change.

  $Rateofchange=\frac{changeiny}{changein\text{ x}}$

Thus, constant rate of change of points $(10,15),(20,20)$ is:

  $\begin{array}{l}Rateofchange=\frac{20-15}{20-10}\\ =\frac{5}{10}\\ =\frac{1}{2}\end{array}$

Hence, rate of change is $\frac{1}{2}$

And, Linear relationships are called proportional if the ration of each non zero $y-value$ compared to corresponding $x-value$ is the same.

For the point $(10,15)$ the ratio is: $\frac{15}{10}=1.5$

And, for the point $(20,20)$ the ratio is: $\frac{20}{20}=1$

And also, for the point $(30,25)$ the ratio is: $\frac{25}{30}=\frac{5}{6}$

Hence, ratio $\frac{yvalue}{\text{x }value}$ is not same for each paired value:

Thus, the linear relationship is not proportional.

Now, consider another example that shows the points: $(2,1000),(4,2000)$ and $(6,3000)$

Thus, constant rate of change is:

  $\begin{array}{l}Rateofchange=\frac{2000-1000}{4-2}\\ =\frac{1000}{2}\\ =500\end{array}$

Hence, rate of change is $500$ .

And, proportional relation is:

For the point $(2,1000)$ the ratio is: $\frac{1000}{2}=500$

And, for the point $(4,2000)$ the ratio is: $\frac{2000}{4}=500$

And also, for the point $(6,3000)$ the ratio is: $\frac{3000}{6}=500$

Hence, ratio $\frac{yvalue}{\text{x }value}$ is same for each paired value.

Thus, the linear relationship is proportional

Therefore, the conclusion is that sometimes a linear relationship that has a constant rate of change also shows proportional relationship.

Conclusion:

Therefore, the conclusion is that sometimes a linear relationship that has a constant rate of change also shows proportional relationship.