Problem 15 - Algebra A Combined Function

Question

Ryan Howard of the Philadelphia Phillies led Major League Baseball in runs batted in for the 2008 regular season. Josh Hamilton of the Texas Rangers, who came in second to Howard, had 16 fewer runs batted in for the 2008 regular season. Together, these two players brought home 276 runs during the 2008 regular season. How many runs batted in each did Howard and Hamilton account for?

Step-by-Step Solution

Answer

Ryan Howard had 146 runs, and Josh Hamilton had 130 runs.

Step 1: Define Variables

Let \( x \) represent the number of runs batted in by Ryan Howard. Since Josh Hamilton had 16 fewer runs, his total runs would be \( x - 16 \).

Step 2: Set Up an Equation

According to the problem, together, Ryan Howard and Josh Hamilton had 276 runs batted in. Therefore, we can set up the equation: \[ x + (x - 16) = 276 \]

Step 3: Simplify the Equation

Combine like terms in the equation from Step 2:\[ 2x - 16 = 276 \]

Step 4: Solve for x

Add 16 to both sides to isolate the term with \( x \):\[ 2x = 292 \]Now, divide both sides by 2 to solve for \( x \):\[ x = 146 \]

Step 5: Determine Runs for Each Player

Since \( x = 146 \), Ryan Howard hit 146 runs. Josh Hamilton had 16 fewer runs, so he had:\[ 146 - 16 = 130 \] runs.

Key Concepts

Defining Variables

In algebra word problems, defining variables is the first and one of the most crucial steps. It helps in translating the problem from words into mathematical expressions. When tackling these types of problems:

• Identify what needs to be found. Here, we are asked to find out the runs batted by both Ryan Howard and Josh Hamilton.
• Assign a variable to the quantity we know the least about. We assigned \( x \) for Ryan Howard's runs because it was directly mentioned.
• Relate other quantities to the defined variable. Josh Hamilton had 16 fewer runs, so his runs are expressed as \( x - 16 \).

By defining variables clearly, you set a strong foundation for the rest of the problem-solving process.

Setting Up Equations

Creating equations from the problem statement is an exciting part of solving word problems. It involves turning the relationships and conditions given in the problem into mathematical statements. For instance:

• You interpret the word 'together' as the sum of the runs scored by both players.
• This condition leads to the equation \( x + (x - 16) = 276 \), since their total combined runs are given as 276.

This equation encompasses all the knowledge extracted from the problem and allows for the application of algebraic methods to solve it.

Simplifying Equations

Once an equation is established, the next step is to simplify it. Simplifying makes solving the equation much easier by reducing it to a more manageable form. Here's how you can simplify:

• Combine like terms. In the equation \( x + (x - 16) = 276 \), you gather the terms involving \( x \). This yields \( 2x - 16 = 276 \).

Simplifying prepares the equation for solving by reducing complexity, making the steps ahead straightforward.

Solving Linear Equations

Solving the simplified linear equation is where the solution emerges. The process involves isolating the variable, thus finding its value. This is done through:

• Arithmetic operations that keep the equation balanced. Adding 16 to both sides of \( 2x - 16 = 276 \) gives \( 2x = 292 \).
• Simplifying further by dividing both sides by 2 results in \( x = 146 \).

Once \( x \) is determined, you can substitute back to find Josh Hamilton’s runs as \( 146 - 16 \), which equals 130. This reflects not only understanding algebra but also interpreting it to provide meaningful results.