Problem 58 - Algebra 2

Question

In a recent season, Monique Currie of the Duke Blue Devils scored 635 points. She made a total of 356 shots, including 3-point field goals, 2-point field goals, and 1-point free throws. She made 76 more 2-point field goals than free throws and 77 more free throws than 3-point field goals. Find the number of each type of shot she made.

Step-by-Step Solution

Answer

Monique made 42 three-pointers, 195 two-pointers, and 119 free throws.

Step 1: Define Variables

Let's assign variables to the different types of shots Monique made. Let \( x \) be the number of 3-point field goals, \( y \) be the number of 2-point field goals, and \( z \) be the number of 1-point free throws she made.

Step 2: Set Up Equations from Total Points and Shots

Using the fact that Monique scored 635 points in total, we can write the equation \( 3x + 2y + z = 635 \). Also, since she made a total of 356 shots, we have the equation \( x + y + z = 356 \).

Step 3: Use Information About Relative Shot Counts

We know that Monique made 76 more 2-point field goals than free throws, so \( y = z + 76 \). Also, she made 77 more free throws than 3-point field goals, therefore \( z = x + 77 \).

Step 4: Substitute to Form System of Equations

Using \( y = z + 76 \) and \( z = x + 77 \), substitute into the equations:1. \( y = (x + 77) + 76 = x + 153 \).2. Substitute into the shot total equation: \( x + (x + 153) + (x + 77) = 356 \).3. Simplify to get \( 3x + 230 = 356 \).

Step 5: Solve for Variables

First, solve \( 3x + 230 = 356 \):\[ 3x = 356 - 230 \ 3x = 126 \ x = 42 \]Then, use \( z = x + 77 \):\[ z = 42 + 77 = 119 \]Finally, use \( y = z + 76 \):\[ y = 119 + 76 = 195 \].

Step 6: Verify the Solution

Checking the point total: \( 3x + 2y + z = 635 \), substitute the values:\( 3(42) + 2(195) + 119 = 126 + 390 + 119 = 635 \).Checking the shot total: \( x + y + z = 356 \), substitute the values:\( 42 + 195 + 119 = 356 \).Both checks are correct, confirming the solution is valid.

Key Concepts

Variables and Expressions

When we begin solving any algebra problem, especially a problem involving different unknown quantities, the first step is to define the variables. For this exercise, where Monique Currie's scores are divided among different types of shots like 3-point, 2-point, and free throws, representing these unknowns with variables makes the problem-solving process clear and structured. We use:

• Variable \( x \) to represent the number of 3-point field goals.
• Variable \( y \) for the 2-point field goals.
• Variable \( z \) for the 1-point free throws.

This naming of variables allows us to translate the word problem into mathematical expressions, enabling systematic calculations. Remember, each variable is an expression representing a specific quantity which is essential in forming equations.

Linear Equations

Once we have our variables defined, the next step is to create linear equations based on the problem's conditions. A linear equation is an equation that graphs a straight line and can be written in the form \( ax + by + cz = d \). In Monique's case:

• We derived the equation \( 3x + 2y + z = 635 \) from the total points scored.
• The equation \( x + y + z = 356 \) comes from the total number of shots made.

Linear equations help specify the constraints clearly. Every equation here represents a condition that the variables must satisfy. This leads us to form a system of equations. Solving these together helps us find the exact values of \( x \), \( y \), and \( z \).

Problem Solving in Algebra

Solving systems of equations is a crucial part of problem solving in algebra. In this exercise, we use substitution, a method that involves replacing one variable based on expressions given by other equations. For instance:

• The information that Monique made 76 more 2-point shots than free throws led to the equation \( y = z + 76 \).
• Knowing she made 77 more free throws than 3-point shots provided \( z = x + 77 \).

By substituting these relations into other equations, our system reduces to a single equation with one variable, \( x \). Solving this gives us \( x = 42 \). We then substitute \( x \) back to find \( z = 119 \) and \( y = 195 \). Once the values are known, it's essential to verify the solution by plugging it back into the original equations, ensuring the calculations are correct and the solution fulfils all the problem's conditions.