Problem 39 - Algebra A Combined Function

Question

During the \(2008-2009\) regular NBA season, the top-scoring player was Dwyane Wade of the Miami Heat. Wade scored a total of 2386 points during the regular season. The number of free throws (each worth one point) he made was 26 less than seven times the number of three-point field goals he made. The number of two-point field goals that Wade made was 176 more than the number of free throws he made. How many free throws, two-point field goals, and three-point field goals did Dwyane Wade make during the \(2008-2009\) NBA season? (Source: National Basketball Association)

Step-by-Step Solution

Answer

Wade made 88 three-pointers, 564 free throws, and 766 two-pointers.

Step 1: Define Variables

Let's define variables to represent the quantities we need to find. Let \( x \) represent the number of three-point field goals, \( y \) represent the number of free throws, and \( z \) represent the number of two-point field goals that Dwyane Wade made during the season.

Step 2: Set Up Equations

According to the problem, we can set up the following equations: 1. Total points scored: \( 3x + y + 2z = 2386 \) (since three-pointers are worth 3 points, free throws are worth 1 point, and two-pointers are worth 2 points).2. The number of free throws: \( y = 7x - 26 \).3. The number of two-point field goals: \( z = y + 176 \).

Step 3: Substitute and Simplify

Substitute \( y = 7x - 26 \) into the third equation: \( z = (7x - 26) + 176 \). Simplify this to obtain \( z = 7x + 150 \).

Step 4: Combine Substituted Equations

Substitute \( y = 7x - 26 \) and \( z = 7x + 150 \) into the total points equation: \[3x + (7x - 26) + 2(7x + 150) = 2386\] This simplifies to a single equation in terms of \( x \).

Step 5: Solve for x

Simplify the combined equation: \[3x + 7x - 26 + 14x + 300 = 2386\] Combine like terms:\[24x + 274 = 2386\]Subtract 274 from both sides: \[24x = 2112\]Divide by 24 to solve for \( x \):\[x = 88\]

Step 6: Solve for y and z

Now that we know \( x = 88 \), substitute back to find \( y \) and \( z \):For \( y \):\[ y = 7(88) - 26 = 590 - 26 = 564 \]For \( z \):\[ z = 7(88) + 150 = 616 + 150 = 766 \]

Step 7: Verification

Verify the values by checking the total points scored: \[3(88) + 564 + 2(766) = 264 + 564 + 1532 = 2386\] The calculated total matches the problem statement, confirming our values are correct.

Key Concepts

Equations

In this word problem, equations play a crucial role in representing the information given in a mathematical form:

The first equation, \( 3x + y + 2z = 2386 \), represents the total points scored by Dwyane Wade. This equation combines all the different scoring methods: three-point field goals, free throws, and two-point field goals.

The second equation, \( y = 7x - 26 \), provides a direct relationship between the number of free throws (\( y \)) and three-point field goals (\( x \)). It shows that Wade scored 26 less free throws than seven times his three-point field goals.

The third equation, \( z = y + 176 \), presents the relationship between the number of two-point field goals (\( z \)) and free throws (\( y \)). It indicates that Wade made 176 more two-point field goals than free throws.

Without these equations, it would be nearly impossible to solve the problem systematically. Each equation translates a part of the word problem into a form that can easily be manipulated and solved. These relationships allow us to piece together the puzzle.

Variable Substitution

Using variable substitution is a powerful method that simplifies solving equations. By substituting one equation into another, we reduce the number of variables and equations:

• First, we use the equation \( y = 7x - 26 \) to express \( y \) in terms of \( x \). This helps us eliminate \( y \) when plugging values into other equations.


• Next, substitute \( y \) in the equation \( z = y + 176 \) to find \( z \) in terms of \( x \). This substitution gives us \( z = 7x + 150 \).

Now, we've expressed all variables in terms of a single variable \( x \).

This method streamlines the solving process as it shifts from handling multiple equations to dealing with a simpler single-variable equation. By reducing complexity through successive substitutions, we make solving the primary equation straightforward.

Problem Solving Steps

Solving word problems in algebra follows a structured process, ensuring all aspects of the problem are addressed. Let's outline these steps:

• **Understand the problem:** Carefully read and comprehend what is being asked. Identify known and unknown variables.


• **Define variables and set up equations:** Assign variables for the unknowns and translate the worded information into algebraic equations, as done with \( x \), \( y \), and \( z \).


• **Substitute and simplify:** Use variable substitution to simplify the equations into a manageable form, focusing on reducing the problem to one or two primary equations with fewer variables.


• **Solve the equations:** Perform algebraic manipulations, like combining like terms and solving linear equations, to find solutions to the variables.


• **Verify:** Plug the solutions back into the original problem to ensure they fulfill all conditions stated in the word problem.

Following these steps provides a logical and organized way to tackle complex problems. It's vital to not only find a solution but also ensure its feasibility in the context of the problem statement.