Problem 21 - Algebra 1

Question

State whether the two quantities have direct variation. You ride your bike at an average speed of 14 miles per hour. The number of miles \(m\) you ride during h hours is modeled by \(m\) = \(14h.\)

Step-by-Step Solution

Answer

Yes, the two quantities (number of miles and hours) have a direct variation as they are proportional to each other with a constant rate of 14.

Step 1: Understanding the relationship

Direct variation means that as one quantity increases, the other quantity increases at a constant rate, or as one decreases, the other decreases at the same rate. In other words, the two quantities are proportional to each other.

Step 2: Identifying the given variables

The two quantities given in the problem are: the number of hours (h) that you ride your bike and the number of miles (m) that you ride. According to the problem, these are related by the equation \(m = 14h\) which implies that for each extra hour you ride, you ride an extra 14 miles.

Step 3: Checking for direct variation

This relationship is an example of direct variation, because the equation can be rewritten in the form \(y = kx\), where \(k\) is the constant of variation. Here, \(m\) corresponds to \(y\), \(h\) corresponds to \(x\), and the constant of variation is 14.

Step 4: Conclusion

Therefore, there is a direct variation between the number of miles you ride and the number of hours you ride your bike. This is because the number of miles you ride (m) is directly proportional to the number of hours (h) you ride, with a constant factor of 14.

Key Concepts

Proportional Relationships

When we talk about proportional relationships in math, we are referring to a scenario where two quantities increase or decrease in tandem at a consistent rate. This means that if one quantity doubles, the other one also doubles, and if one is cut in half, so is the other. In the context of your problem, where biking hours and miles are involved, as the hours increase, the miles do so at a rate that is always the same—proportionally.

For instance, if you ride your bike for 2 hours at an average speed of 14 miles per hour, you will cover 28 miles. If your riding time increases to 4 hours, the distance doubles to 56 miles. This keeps true no matter how long you ride because the relationship between time and distance is constant; the more you ride, the more miles you cover in direct proportion to the time spent riding.

Constant Rate of Change

The term constant rate of change is a key attribute of direct variation. It is the unchanging value that relates the changes in one quantity to the changes in another. In algebra, when we plot a graph representing a direct variation, the constant rate of change corresponds to the slope of the line.

In your textbook exercise, this constant is 14, representing the 14 miles you travel for every hour of bike riding. Mathematically, it can be depicted as the equation, \( m = 14h \), highlighting that the 'miles' (m) changes by 14 for every one-unit increase in 'hours' (h). Hence, the graph of this equation is a straight line going through the origin with a slope (or rate of change) of 14, which demonstrates that the relationship between time and miles is both direct and constant.

Variables in Algebra

Let's dig into the notion of variables in algebra. Variables are symbols like 'x', 'y', 'm', 'h', etc., that represent numbers in equations. They are the core components of algebraic expressions and can stand for unknown values or quantities that are able to change. When we establish relationships between variables, we are often trying to understand how the change in one affects the other.

In the biking scenario, 'h' (hours) and 'm' (miles) are variables. The equation \( m = 14h \) helps us to predict how many miles will be covered given a certain number of hours 'h'. The beauty of algebra is its ability to generalize. Here, the number '14' is constant, but 'm' and 'h' can vary. You could ride for 5 hours, 10 hours, or any number of hours, and this equation will still tell you the corresponding distance. This flexibility is what makes algebra such a powerful tool in mathematics.