Problem 4 - College Algebra

Question

If a man sawed a 10 -ft oard into two sections and one section was \(n \mathrm{ft} \mathrm{l} \mathrm{ng},\) how long would the other section be in terms of \(n\) ?

Step-by-Step Solution

Answer

The other section is \(10 - n\) feet long.

Step 1: Understanding the Problem

The problem states that a 10-foot board is cut into two sections. We are asked to express the length of one section in terms of the given length of the other section, denoted by \(n\).

Step 2: Setting Up the Equation

Since the total length of the board is 10 feet, we need to establish an equation where the sum of the two sections equals 10. Let's set one section as \(n\), thus we can say: \(n + x = 10\), where \(x\) is the length of the other section.

Step 3: Isolating the Unknown Length

To find \(x\), we need to isolate it in the equation. We do this by subtracting \(n\) from both sides of the equation: \(x = 10 - n\).

Step 4: Conclusion

The length of the other section in terms of \(n\) is \(10 - n\).

Key Concepts

Problem-Solving

When it comes to solving algebraic problems, understanding the scenario is crucial. In this situation, the problem informs us about a 10-foot board that is split into two sections. The key task is to find the length of the second section using the given measurement of the first section, denoted as \( n \). This is a classic case of problem-solving in algebra, where identifying what we know and what we need to find is essential. By clearly recognizing the total length and the given length of one part, we can proceed to solve for the unknown component. Always remember to:

• Comprehend what is being asked.
• Identify known values.
• Recognize what needs to be calculated.

These steps pave the way for setting up the right equation and finding a solution efficiently.

Equation Setup

Setting up an equation is about translating the problem into a mathematical form. In this exercise, you are provided with a 10-foot board and a section of it that is \( n \) feet long. We need to establish a relationship that represents the total length of the board using both sections. This can be represented as:

• Let \( n \) be the length of the known section.
• Let \( x \) be the length of the unknown section.
• The equation will be \( n + x = 10 \).

This equation captures the entire scenario — it shows that the sum of the two sections' lengths equals the length of the board. Proper equation setup is crucial for problem-solving in algebra because it forms the foundation upon which you can apply mathematical operations to find unknown values.

Isolation of Variables

Once our equation is set up, the next challenge is to find the value of the unknown variable — in this scenario, \( x \). This process is known as isolation of variables, and it involves manipulating the equation to solve for the desired variable. With the equation \( n + x = 10 \), you want to isolate \( x \). You can do this by:

• Subtracting \( n \) from both sides of the equation.
• This gives \( x = 10 - n \).

By isolating \( x \), you express the unknown section's length purely in terms of the known variable \( n \). Mastering isolation of variables is a valuable algebraic skill, allowing you to solve for unknowns quickly and efficiently. It’s simply about performing operations that eliminate other terms, providing a clear value for your variable of interest.