Problem 40 - Algebra A Combined Function

Question

For \(2009,\) the WNBA's top scorer was Cappie Poindexter of the Phoenix Mercury. She scored a total of 648 points during the regular season. The number of two-point field goals that Poindexter made was 22 fewer than five times the number of three-point field goals she made. The number of free throws (each worth one point) she made was 60 fewer than the number of two- point field goals she made. Find how many field goals, three-point field goals, and free throws Cappie Poindexter made during the 2009 regular season. (Source: Women's National Basketball Association)

Step-by-Step Solution

Answer

Two-point FG: 193, Three-point FG: 43, Free throws: 133.

Step 1: Define Variables

Let \( x \) represent the number of two-point field goals, \( y \) represent the number of three-point field goals, and \( z \) represent the number of free throws Poindexter made.

Step 2: Write Equations Based on the Problem Statement

1. The equation reflecting the relationship between two-point and three-point field goals is \( x = 5y - 22 \).2. The equation for the free throws related to two-point field goals is \( z = x - 60 \).3. The total points scored are represented by the equation \( 2x + 3y + z = 648 \).

Step 3: Substitute and Simplify

Substitute \( x = 5y - 22 \) into the third equation: \( 2(5y - 22) + 3y + z = 648 \). Simplify this to \( 10y - 44 + 3y + z = 648 \).

Step 4: Use Equation for Free Throws

Substitute the expression for free throws \( z = x - 60 \) (or \( z = 5y - 82 \) from substituting for \( x \) into it) into the equation: \( 13y - 44 + 5y - 82 = 648 \). Simplify to get \( 18y - 126 = 648 \).

Step 5: Solve for Three-Point Field Goals

Solving the simplified equation \( 18y - 126 = 648 \), we obtain \( 18y = 774 \). Thus, \( y = 43 \).

Step 6: Find the Number of Two-Point Field Goals

Use the equation \( x = 5y - 22 \) to find \( x \). Substitute \( y = 43 \), we get \( x = 5(43) - 22 = 193 \).

Step 7: Find the Number of Free Throws

From \( z = x - 60 \), with \( x = 193 \), we find \( z = 193 - 60 = 133 \).

Step 8: Verify the Solution

Substitute \( x = 193 \), \( y = 43 \), and \( z = 133 \) back into the total points equation: \( 2(193) + 3(43) + 133 = 648 \). Simplifying gives \( 386 + 129 + 133 = 648 \), confirming the solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements formulated with variables, constants, and arithmetic operations. They show the relationship between different quantities. In our exercise, three key equations are crucial to solving the problem. These equations reflect:

• The relationship between the number of two-point field goals and three-point field goals: \( x = 5y - 22 \)
• The number of free throws in relation to two-point field goals: \( z = x - 60 \)
• The total points scored through field goals and free throws: \( 2x + 3y + z = 648 \)

Each equation captures the connections and constraints as detailed in the problem description. Understanding how to set up such equations is essential in solving algebraic problems, as they represent how quantities depend on one another. This is the foundation of what we're trying to determine.

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves expressing one variable in terms of another and substituting it into another equation. This method simplifies the system and allows us to solve for one variable at a time. Here is how it was applied in our problem:

• We initially defined \( x \) in terms of \( y \) from the first equation: \( x = 5y - 22 \).
• We then replaced \( x \) in the equation for the total points, giving us a simplified expression in \( y \) and \( z \): substituting into \( 2x + 3y + z = 648 \).
• This lets us manage the system by reducing it to fewer variables and equations.
• Finally, free throws \( z \) were expressed in terms of \( y \), completing the necessary substitutions needed to find each variable.

By using substitution effectively, we reduce the complexity of the problem, peeling it down to simpler terms that can be handled more easily.

Problem Solving Steps

Problem-solving involves a defined sequence of steps that lead to a solution. These steps help in methodically handling algebraic problems. In this case, here's how each step guided us towards the solution:

• Define Variables: We began by clearly laying out what each variable represented: \( x \) for two-point goals, \( y \) for three-point goals, and \( z \) for free throws.
• Write Equations: With variables in place, we translated the problem's conditions into algebraic expressions.
• Substitute & Simplify: We used substitution to express all equations in terms of a single variable, which involved a series of substitutions and simplifications.
• Solve: We solved the final simplified equation to find \( y = 43 \).
• Back-substitute: Using the value of \( y \), we found \( x = 193 \) and \( z = 133 \).
• Verify: Finally, the solution was verified by checking that the original equations were satisfied with these values.

Each step is detailed and vital, ensuring no part of the process is overlooked, leading to a correct and verified solution.