Problem 27 - Algebra 1: Concepts and Skills

Question

Use the following information. A basketball team scored 102 points in a playoff game. Each field goal is 2 points and each free throw is 1 point. The team scored no 3 -point field goals. Write a linear model for the number of points the team scored in terms of field goals \(x\) and free throws \(y .\)

Step-by-Step Solution

Answer

The linear model in terms of field goals \(x\) and free throws \(y\) for the team's score of 102 points is: \(2x + y = 102\).

Step 1: Understanding the problem

First, establish that you understand the relationship between the variables in question. In this case, two types of scores contribute to the total points. Each field goal (x) represents 2 points and each free throw (y) represents 1 point. This information gives us the weighting in our model.

Step 2: Formulating the linear model

A linear model can be represented by a linear equation in two variables. In this case, we want to represent the team's total score (102 points) in terms of field goals and free throws. Since the worth of field goals (x) is 2 points and free throws (y) are 1 point, our linear model is given as: \(2x + y = 102\), where x is the number of field goals and y is the number of free throws.

Step 3: Verifying the model

Finally, it is important to verify that the model you have is correct. It represents the correct relationship between field goals, free throws, and total points. The equation \(2x + y = 102\) means that the total of twice the number of field goals plus the number of free throws will always be equal to the total score (102). If you substitute some values for x and y into the equation, you will see that it holds true, hence validating the correctness of the model.

Key Concepts

Linear Models

In the context of algebra, a linear model is a mathematical equation used to represent a relationship between two variables, usually in the form of a straight line. Linear models are incredibly useful because they allow us to describe and predict outcomes based on given variables.
For example, to represent the points scored by a basketball team, we can create a linear equation. This equation considers different scoring methods like field goals and free throws. Each field goal contributes 2 points, while each free throw contributes 1 point.
Therefore, if a team needs to account for its total points, creating a linear model helps us see how different combinations of the number of field goals and free throws sum up to their overall score.

• In our example, the equation is: \(2x + y = 102\).
• Here, \(x\) represents the number of field goals.
• \(y\) represents the number of free throws.

With linear models, understanding the interplay between different variables becomes easier, allowing for clear and concise predictions.

Algebra

Algebra is a branch of mathematics dealing with variables and the rules to manipulate these symbols. It lays the foundation for discovering unknowns and understanding relationships between figures. Such usage is evident in forming linear equations, like our basketball scoring problem.
Algebra uses letters to stand in for numbers in equations and formulas. This approach makes it easier to solve problems and explore relationships between different quantities without needing numeric values initially.

• In the case of the basketball team, the equation \(2x + y = 102\) is set up through algebra.
• The '2' and '1' act as coefficients for the respective variables \(x\) and \(y\).
• These coefficients represent the point values of field goals and free throws.

By setting up algebraic equations, we can replace the variables with suitable numbers to verify equation solutions, as demonstrated by substituting values into the team score equation.

Basketball Scoring

Basketball scoring encompasses various ways that players can earn points during a game. Each method has a different point value:

• A field goal typically adds 2 points to the team's score.
• A free throw usually grants 1 point.
• Three-point field goals are worth 3 points, though not considered in our scenario.

Understanding these scoring methods is vital for building the team's point total. As reflected in our linear equation \(2x + y = 102\), various combinations of field goals and free throws result in the 102 total points achieved by the team.
By grasping these basketball scoring rules, players, coaches, and fans gain insights into game strategy: knowing how points are accumulated helps teams plan their actions and maximize overall performance. In this exercise, we specifically calculate how various mixes of field goals and free throws sum to the team's final score.