Problem 55 - Algebra A Combined Function

Question

Write each algebraic expression described. A 10 -foot board is cut into two pieces. If one piece is \(x\) feet long, express the other length in terms of \(x\).

Step-by-Step Solution

Answer

The other piece is \(10 - x\) feet long.

Step 1: Understand the Problem

You have a board that is 10 feet long and it's being divided into two pieces. One piece is given as being \(x\) feet long. You need to find the length of the other piece.

Step 2: Set Up the Equation

Write an equation representing the length of the board. The board's total length is 10 feet, and it is separated into two parts: \(x\) and the other piece. The sum of these two parts must equal 10 feet. Therefore, the equation is: \[ x + \text{other piece} = 10 \]

Step 3: Solve for the Other Piece

Rearrange the equation from Step 2 to solve for the length of the other piece. Subtract \(x\) from 10: \[ \text{other piece} = 10 - x \]

Step 4: Express the Other Piece in Terms of x

Finally, express the length of the other piece using the variable \(x\). The algebraic expression for the length of the other piece is: \(10 - x\).

Key Concepts

Understanding Equations

Equations are mathematical sentences that use an equal sign to show that two expressions are equal. In this context, we are using them to describe how a whole, like a board, can be broken down into smaller parts.
When you write an equation, you are representing a real-world scenario using numbers and variables. This allows you to reason through a problem systematically.
For example, when a 10-foot board is cut into two pieces, you can express the idea that the board is split into two lengths using an equation:

• 10 = a + x

Here, the entire length of the board (10 feet) is equal to the sum of the two pieces ("a" and "x"). Through this equation, you can solve for any unknown part, given some other information. This approach simplifies problem-solving and enhances clarity.

The Role of Variable Substitution

Variable substitution is the process of replacing variables with numbers or other expressions to simplify an equation or solve it. This technique is crucial when dealing with algebraic expressions.
In our exercise, you are already given that one of the lengths of the board is represented by the variable "x". Your goal is to express the unknown part of the board in terms of "x".
By using variable substitution, you can express unknown quantities in terms of known quantities. For instance, starting with the equation derived from the board problem:

• 10 = x + ext{other piece}

Substitute the length variable for the unknown piece, leading to the equation rearrangement: \( \text{other piece} = 10 - x \). This substitution makes it easier to understand the length relationship between parts of the board.

Effective Problem Solving Steps

Solving algebraic problems effectively involves breaking down the process into clear, manageable steps. Here's how you achieve that with our board problem:

• Understand the Problem: Recognize what is known and what needs to be found. Here, a 10-foot board is split, with one part's length given as "x".
• Set Up the Equation: Use the total length to form an equation. You know the sum of both parts equals the length of the board, which leads you to \( 10 = x + \text{other piece} \).
• Solve for the Unknown: Rearrange the equation to isolate the unknown quantity on one side. This process leads to \( \text{other piece} = 10 - x \).
• Express the Solution: Write the solution in a clear algebraic form. The length of the other piece is expressed as \( 10 - x \).

By following these steps, problem-solving becomes more systematic and less overwhelming. Always ensure each step is understood and justified to build confidence in handling algebraic expressions.