Problem 16 - Algebra 2

Question

BASEBALL During the 2005 season, Jacque Jones and Matthew LeCroy of the Minnesota Twins hit a combined total of 40 home runs. Jones hit 6 more home runs than LeCroy. How many home runs did each player hit? Define a variable, write an equation, and solve the problem.

Step-by-Step Solution

Answer

Matthew LeCroy hit 17 home runs, Jacque Jones hit 23 home runs.

Step 1: Define the Variable

Let's define a variable to represent one of the unknown quantities. Let \( x \) be the number of home runs hit by Matthew LeCroy during the 2005 season.

Step 2: Express the Other Variable Using Defined Variable

Given that Jacque Jones hit 6 more home runs than Matthew LeCroy, we can write Jacque Jones's home runs as \( x + 6 \).

Step 3: Set Up the Equation

Based on the problem, we know the total number of home runs hit by both players is 40. Thus, we can set up the equation: \[ x + (x + 6) = 40 \]

Step 4: Simplify the Equation

Combine like terms to simplify the equation:\[ 2x + 6 = 40 \]

Step 5: Solve for the Variable

Subtract 6 from both sides of the equation to isolate terms with \( x \):\[ 2x = 34 \]Now, divide both sides by 2:\[ x = 17 \]

Step 6: Find the Number of Home Runs for Jacque Jones

Since we defined \( x \) as the number of home runs by Matthew LeCroy, Jacque Jones hit \( x + 6 \) home runs:\[ 17 + 6 = 23 \]So, Jacque Jones hit 23 home runs.

Key Concepts

Variable Definition

Understanding variables is crucial in solving algebraic equations. A variable is a symbol, often a letter, that represents an unknown number in a mathematical expression or equation. In our baseball exercise, we were tasked with finding the number of home runs hit by two players. We defined the variable \( x \) to represent the number of home runs hit by Matthew LeCroy.
This decision was made because we often start by assigning variables to the lesser-known quantity to simplify our calculations. Once the variable for LeCroy was established, it allowed us to express Jacque Jones's home runs as \( x + 6 \), since Jones hit 6 more than LeCroy. By defining the variable first, we set the foundation to build our equation later on.

Solving Linear Equations

Linear equations are equations where the highest power of the variable is one. They are called 'linear' because they graph as straight lines. In our problem, we set up a linear equation to find out how many home runs each player hit.

• We formed the equation \( x + (x + 6) = 40 \) based on the information given that their total home runs were 40.
• We then combined like terms to get \( 2x + 6 = 40 \).
• To simplify, we subtracted 6 from both sides, resulting in the equation \( 2x = 34 \).
• Finally, by dividing both sides by 2, we solved for \( x \) to find \( x = 17 \).

These steps involve common techniques in solving linear equations such as simplifying and isolating the variable, which are key skills in algebra.

Problem-Solving Skills

Improving problem-solving skills requires practice and understanding of each step in the process. Let's break down how we applied these skills to our exercise:

• Understand the Problem: Clearly read and comprehend what is being asked. We identified that we needed to find the home runs for both players.
• Define Variables: As seen, defining a variable right at the beginning helps in systematically approaching the solution.
• Equation Setup: Translate the word problem into a mathematical equation. This requires an understanding of relationships, like knowing Jones hits 6 more home runs than LeCroy.
• Solve the Equation: Use algebraic techniques to find the value of the variable, ensuring each step follows logically from the one before.
• Check Your Work: Once a solution is found, verify it by substituting back into the context to ensure it fits all parts of the problem.

By consistently applying these strategies, students can enhance their problem-solving skills and tackle similar algebra problems confidently.