Explanation: In the context of linear algebra, we often represent real-world problems using linear equations. The problem at hand involves the cost of items, specifically balls and bats. To solve this, we need to translate the given information into a mathematical equation.
Let's define the variables:
- Let \( x \) be the cost of one ball.
- Let \( y \) be the cost of one bat.
The problem states that the total cost of 2 balls and 1 bat is 100 units. This can be expressed as a linear equation:
\[ 2x + y = 100 \]
Here's a step-by-step breakdown of how we arrived at this equation:
1. **Identify the variables**: We assign \( x \) to the cost of one ball and \( y \) to the cost of one bat.
2. **Formulate the equation**: The total cost of 2 balls is \( 2x \) and the cost of 1 bat is \( y \). Adding these together gives the total cost, which is 100 units.
3. **Write the equation**: Combining the terms, we get \( 2x + y = 100 \).
This equation represents the relationship between the cost of the balls and the bat. It is a linear equation in two variables, \( x \) and \( y \), and it describes the total cost of the items.
To further understand the concept, let's consider why the other options are incorrect:
- **Option B (2x + 2y = 100)**: This equation suggests that the total cost of 2 balls and 2 bats is 100 units, which is not what the problem states. The problem specifies the cost of 2 balls and 1 bat.
- **Option C (2x + y = 200)**: This equation suggests that the total cost of 2 balls and 1 bat is 200 units, which is twice the given total cost. The problem states the total cost is 100 units.
- **Option D (x + y = 100)**: This equation suggests that the total cost of 1 ball and 1 bat is 100 units, which does not match the given information about 2 balls and 1 bat.
In summary, the correct representation of the problem in linear algebra is \( 2x + y = 100 \). This equation accurately reflects the given information and can be used to solve for the individual costs of the ball and the bat if additional information is provided.