Problem 16 - Algebra A Combined Function

Question

The highest scorer during the WNBA 2008 regular season was Diana Taurasi of the Phoenix Mercury. Over the season, Taurasi scored 141 more points than the second-highest scorer, Cappie Pondexter, also of the Phoenix Mercury. Together, Taurasi and Pondexter scored 1499 points during the 2008 regular season. How many points did each player score over the course of the season?

Step-by-Step Solution

Answer

Pondexter scored 679 points, and Taurasi scored 820 points.

Step 1: Understand the problem

We need to find the individual points scored by Diana Taurasi and Cappie Pondexter during the 2008 WNBA regular season. We know that Taurasi scored 141 more points than Pondexter and that together they scored 1499 points.

Step 2: Set up the equations

Let \( x \) be the points scored by Cappie Pondexter. Since Diana Taurasi scored 141 more points than Pondexter, we can express her points as \( x + 141 \). The sum of their scores is given as 1499 points, so we establish the equation: \( x + (x + 141) = 1499 \).

Step 3: Simplify and solve the equation

Combine like terms in the equation: \( 2x + 141 = 1499 \). Next, isolate \( x \) by subtracting 141 from both sides to get \( 2x = 1358 \). Finally, divide both sides by 2 to find \( x = 679 \).

Step 4: Find the points for each player

With \( x = 679 \), we know Cappie Pondexter scored 679 points. To find Diana Taurasi's score, use the expression \( x + 141 \): 679 + 141 = 820. Therefore, Diana Taurasi scored 820 points.

Key Concepts

Linear equations

Linear equations are a fundamental part of algebra, often introduced early in mathematics education. They involve expressions where each term is either a constant or the product of a constant and a single variable.

In this problem, the linear equation helps model the relationship between the points scored by Diana Taurasi and Cappie Pondexter. The equation is set up based on the information given:

• Let \( x \) represent the points scored by Cappie Pondexter.
• Diana Taurasi scored \( x + 141 \) points.
• The total score for both players is 1499.

By understanding these relationships, we form the linear equation: \[x + (x + 141) = 1499\]Simplifying this equation allows us to find the values of \( x \), which represent the individual scores of the players. Understanding how to set up and solve linear equations like this is crucial for solving real-world problems effectively.

Problem solving

Problem solving with algebraic expressions involves several key steps: understanding the problem, setting up a mathematical model, simplifying the model, and solving the equations.

In the context of the exercise, the problem can initially seem daunting due to the narrative form of the information. To combat this:

• Comprehend the problem statement. Identify what is known, which is that Taurasi scored 141 more points than Pondexter, and both together scored 1499 points.
• Translate this understanding into algebraic expressions and equations.
• Simplify the resulting expression to make it solvable. In this exercise, simplification is achieved by combining similar terms, subtracting constants from both sides, and dividing both sides of the equation to isolate the variable of interest.

Practicing these problem-solving steps can enhance your mathematical capabilities, making it easier to tackle algebraic problems in exams or real-life situations.

System of equations

Systems of equations are collections of equations that are solved together because they share variables. While this particular problem involves only one equation to isolate two unknowns, understanding systems of equations is still essential.

In scenarios where more than one equation is necessary, systems of equations become invaluable, even when derived from word problems.

• Each equation corresponds to different constraints or conditions described in the problem.
• Finding solutions involves methods such as substitution, elimination, or using matrices (though matrices are not covered in this simple exercise).

For this problem, only one linear equation sufficed due to the simplicity of the constraints. However, learning to handle more complex problems with multiple constraints can prepare you for tackling advanced topics in mathematics and real-world issues requiring a layered approach.