1. Equations & Inequalities

Linear Equations

1. Equations & Inequalities

Linear Equations

3:32 minutes

Problem 16

Textbook Question

Solve each problem. (Modeling) Online Retail SalesProjected retail e-commerce sales (in billions of dollars) for the years 2016–2022 can be modeled by the equation y=52.304x+396.80, where x=0 corresponds to 2016, x=1 corresponds to 2017, and so on. Based on this model, find projected retail e-commerce sales in 2022 to the nearest tenth of a billion. (Data from www.statista.com)

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Identify the given linear model for projected retail e-commerce sales: $y = 52.304x + 396.80$, where $x$ represents the number of years since 2016.

Determine the value of $x$ that corresponds to the year 2022. Since $x=0$ corresponds to 2016, calculate $x$ as \$2022 - 2016$.

Substitute the value of $x$ found in the previous step into the equation $y = 52.304x + 396.80$ to find the projected sales for 2022.

Perform the multiplication and addition operations in the equation to express $y$ in terms of billions of dollars.

Round the resulting value of $y$ to the nearest tenth of a billion to get the final projected retail e-commerce sales for 2022.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Functions and Models

A linear function models a relationship between two variables with a constant rate of change, represented by y = mx + b. Here, y represents the sales, x the year index, m the slope indicating yearly increase, and b the initial value at x=0. Understanding this helps interpret and use the given equation to predict values.

Interpreting Variables and Parameters

In the equation y = 52.304x + 396.80, x is the number of years since 2016, and y is the projected sales in billions. Recognizing how x corresponds to specific years allows substitution to find sales for any given year, such as 2022, by calculating x = 6.

Evaluating Functions for Specific Inputs

To find the projected sales for 2022, substitute x = 6 into the linear model and compute y. This process involves basic arithmetic and understanding function evaluation, which is essential for applying models to real-world data and making predictions.

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