Problem 14 - Algebra A Combined Function

Question

In the 2008 Summer Olympics, the U.S. team won 13 more gold medals than the Russian team. If the total number of gold medals won by both teams was \(59,\) find the number of gold medals won by each team. (Source: Beijing 2008 Olympic Games)

Step-by-Step Solution

Answer

The Russian team won 23 gold medals and the U.S. team won 36 gold medals.

Step 1: Define Variables

Let's define variables to represent the number of gold medals each team won. Let \( x \) be the number of gold medals won by the Russian team, and \( y \) be the number of gold medals won by the U.S. team.

Step 2: Set Up Equations

We know from the problem statement:1. The U.S. team won 13 more gold medals than the Russian team. This gives us the equation: \( y = x + 13 \).2. The total number of gold medals won by both teams was 59. This gives us the equation: \( x + y = 59 \).

Step 3: Substitute and Solve for One Variable

Substitute \( y = x + 13 \) into the equation \( x + y = 59 \):\[ x + (x + 13) = 59 \]Simplify and solve for \( x \):\[ 2x + 13 = 59 \]Subtract 13 from both sides:\[ 2x = 46 \]Divide both sides by 2:\[ x = 23 \]

Step 4: Solve for the Other Variable

Now that we have \( x = 23 \), use the equation \( y = x + 13 \) to find \( y \):\[ y = 23 + 13 \]\[ y = 36 \]

Step 5: Verify the Solution

Check that the solution meets the conditions:- The U.S. team has 13 more gold medals than the Russian team: \( 36 = 23 + 13 \), which is correct.- The total number of medals is 59: \( 23 + 36 = 59 \), which is also correct.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that can help us solve a wide range of problems, including word problems like the one given in the exercise. In a linear equation, the highest power of the variable is one, which means they form straight-line graphs when plotted. These equations usually take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
When dealing with word problems, linear equations allow us to express relationships between quantities conveniently. For instance, in our problem, the relationship between the gold medals won by the U.S. and Russian teams can be expressed using linear equations. Creating these equations from the word problem is an important skill that helps transform the given information into a form that can be manipulated mathematically.

Variable Definition

Defining variables is an essential first step in solving algebraic word problems. A variable is a symbol, often a letter, that represents an unknown quantity we are trying to find. In the original problem, we define:

• \( x \) as the number of gold medals won by the Russian team.
• \( y \) as the number of gold medals won by the U.S. team.

By defining these variables, we set the stage for setting up equations that represent the relationships described in the problem statement. Clear variable definition is crucial as it can help avoid confusion and make sure each step of the solution is understandable. It's like setting up placeholders for information we need to uncover as we proceed with our solution.

Equation Substitution

Equation substitution is a powerful technique in algebra used to solve systems of equations. Once we have two equations representing the problem—one for the relation between the teams' medals and another for the total medals—we can substitute one equation into the other to find the value of one variable.
In this exercise, we set up the equations:

• \( y = x + 13 \)
• \( x + y = 59 \)

We substitute \( y = x + 13 \) into \( x + y = 59 \) to reduce the system to a single equation with one variable.
By making substitution, we transform a problem with two unknowns into one we can solve easily. This significantly simplifies our problem, allowing us to solve for one variable first and then use it to find the second variable.

Problem Solving Steps

Solving algebraic word problems involves several methodical steps that help break down the problem and achieve the correct solution.

To solve:

• **Define Variables:** Assign variables to represent the unknowns.
• **Set Up Equations:** Use the information in the problem to create equations. For example, we know the U.S. won more medals, so \( y = x + 13 \).
• **Substitute and Solve:** Substitute known relationships into the equations to find the value of one variable.
• **Verify Solution:** Check your solution against the conditions given in the problem to ensure they are satisfied.

Understanding and following these structured steps not only help solve the current problem but also instills good problem-solving habits that are applicable to other algebraic challenges. They guide you to logically progress from unknowns to knowns, ensuring clarity and accuracy.