Question

In the equation \(y=5 x+3, y\) represents the total cost of a trip to the aquarium for \(x\) people in a car. Name the slope and \(y\)-intercept of the graph of the equation and explain what each value represents.

Short answer

Slope: 5, representing extra cost per person. Y-intercept: 3, representing the base cost.

Step-by-step solution

Identify Components of the Equation

The given equation is in the linear form: \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept. The equation given is \(y = 5x + 3\). Thus, we identify the slope \(m\) as 5 and the y-intercept \(b\) as 3.

Interpret the Slope

The slope \(m = 5\) indicates how much the total cost \(y\) increases for each additional person \(x\) joining the trip. Therefore, the slope of 5 means that the cost increases by 5 units (e.g., dollars) for each extra person.

Interpret the Y-Intercept

The y-intercept \(b = 3\) represents the fixed starting cost when \(x = 0\). This means when no one joins the trip (except the driver, if implicitly considered), the cost is 3 units (e.g., dollars). Thus, this might cover some base charges independent of the number of people.

Conclusion About the Equation

In conclusion, the equation tells us that for each person added to the trip, the total cost increases by 5, starting from a base cost of 3 when no additional people are counted.

Key concepts

Slope

In the world of linear equations, the slope is a critical component that helps us understand how variables are related. When you look at the equation form of a line, which is typically written as \(y = mx + b\), the \(m\) represents the slope. But what exactly is slope?

The slope of a line is a measure of its steepness and direction. In our equation \(y = 5x + 3\), the slope is \(5\). This means that for every one unit increase in \(x\), which represents the number of people in this case, the cost \(y\) increases by 5 units.

• The slope is often described as "rise over run," indicating how much \(y\) (the rise) changes as \(x\) (the run) changes.
• A positive slope, like our 5, shows that as the number of people increases, the cost also increases, reflecting a direct relationship.

Therefore, in practical terms, the slope gives us information about how costs increase with the number of attendees heading to the aquarium.

Y-Intercept

The y-intercept is another essential component of the linear equation and plays a significant role in understanding fixed costs, particularly in financial contexts like this one. In the equation \(y = mx + b\), the \(b\) stands for the y-intercept.

In our specific equation \(y = 5x + 3\), the y-intercept is \(3\). But what does this mean for our scenario? The y-intercept is the point where the line crosses the y-axis, which corresponds to the situation where \(x = 0\).

• In this aquarium trip problem, it represents the initial or starting cost when 0 people are involved apart from the essential driver, if any.
• This value of 3 can be thought of as a fixed cost or a base fee that’s independent of the number of additional trip participants.

This base fee is crucial as it provides a starting point before the variable cost based on the number of people kicks in.

Cost Analysis

Doing a cost analysis with a linear equation helps visualize and compute the expected costs in practical situations like event planning. Given the equation \(y = 5x + 3\), we can understand how total costs will behave as more people join the trip.

Cost analysis with linear equations often involves:

• Identifying variable costs, which are dependent on some changing factor, in this case, the number of people \(x\). For every person, the cost increases by a factor of 5, denoting the slope.
• Recognizing fixed costs, highlighted by the y-intercept, which is \(3\) here. This fixed cost does not change with the number of people.

By using cost analysis, you can make budget forecasts, decide on pricing structures, or even optimize group sizes for cost efficiency.

Understanding both the slope and y-intercept in this linear model is vital, as they allow you to break down the total cost into understandable parts—variable and fixed costs—thereby aiding in effective financial planning.