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Chapter 5: Problem 39

A basketball player scored 26 points in one game. In basketball, some baskets are worth 3 points, some are worth 2 points, and free-throws are worth 1 point. \(\mathrm{He}\) scored four more 2 -point baskets than he did 3 -point baskets. The number of free-throws equaled the sum of the number of 2 -point and 3 -point shots made. How many free-throws, 2 -point shots, and 3 -point shots did he make?

Short Answer

Expert verified
The player made 2 3-point shots, 6 2-point shots, and 8 free-throws.

Step by step solution

01

- Define Variables

Let \( x \) be the number of 3-point baskets, \( y \) be the number of 2-point baskets, and \( z \) be the number of free-throws.
02

- Write Equations Based on Problem Description

From the problem, derive the system of equations:1. Total points scored: \( 3x + 2y + z = 26 \).2. Four more 2-point baskets than 3-point baskets: \( y = x + 4 \).3. Number of free-throws equals the sum of the number of 2-point and 3-point shots: \( z = x + y \).
03

- Substitute \( y \) Into the Equations

Substitute \( y = x + 4 \) into the first and third equations:1. \( 3x + 2(x + 4) + z = 26 \).2. \( z = x + (x + 4) \).
04

- Simplify and Solve for \( z \)

Simplify the second substituted equation to find \( z \):1. \( z = x + x + 4 = 2x + 4 \).
05

- Substitute \( z \) and Simplify

Substitute \( z = 2x + 4 \) back into the first equation:1. \( 3x + 2(x + 4) + (2x + 4) = 26 \).2. Simplify: \( 3x + 2x + 8 + 2x + 4 = 26 \).3. Combine like terms: \( 7x + 12 = 26 \).
06

- Solve for \( x \)

Isolate \( x \):1. \( 7x = 14 \).2. \( x = 2 \).
07

- Solve for \( y \) and \( z \)

Use \( x \) to find \( y \) and \( z \):1. \( y = x + 4 = 2 + 4 = 6 \).2. \( z = 2x + 4 = 2(2) + 4 = 8 \).
08

- Verify the Answers

Verify the numbers of baskets:1. Points from 3-point baskets: \( 3 \times 2 = 6 \).2. Points from 2-point baskets: \( 2 \times 6 = 12 \).3. Points from free-throws: \( 1 \times 8 = 8 \).4. Total points: \( 6 + 12 + 8 = 26 \), which matches the given total points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

system of equations
A system of equations is a set of two or more equations that share the same variables. In this exercise, we have three variables: the number of 3-point baskets, 2-point baskets, and free-throws. These are represented as \( x \), \( y \), and \( z \), respectively. To solve a system of equations, we need to find values for these variables that satisfy all the equations simultaneously.
This often involves combining the equations in various ways to isolate and solve for the variables.
substitution method
The substitution method is a technique used to solve systems of equations. We start by solving one of the equations for one variable and then substitute that expression into another equation.
In this problem, the second equation tells us that the number of 2-point baskets, \( y \), is four more than the number of 3-point baskets, \( x \). We can write this as \( y = x + 4 \).
By substituting this expression into the other equations, we can reduce the number of variables and thus simplify the problem. This makes it easier to solve for the remaining variables.
defining variables
Before solving a problem, clearly defining variables is crucial. It helps avoid confusion and ensures that each part of the problem is addressed appropriately.
In this exercise, we defined:
  • \( x \), the number of 3-point baskets
  • \( y \), the number of 2-point baskets
  • \( z \), the number of free-throws
These definitions allow us to translate the word problem into mathematical equations that can be systematically solved.
solving equations
After substituting the known relationships between variables into the equations, we simplify and solve for the unknowns.
For example, by substituting \( y = x + 4 \) and \( z = 2x + 4 \) back into the first equation, we transformed it into an equation with one variable:
\[ 7x + 12 = 26 \].
We solved this by isolating \( x \), giving us \( x = 2 \).
After finding \( x \), we used it to find \( y \) and \( z \):
  • \( y = x + 4 \Rightarrow y = 6 \)
  • \( z = 2x + 4 \Rightarrow z = 8 \)
This sequential approach guarantees that all variables are correctly determined, verifying the total points scored to ensure the solution's accuracy.

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