Solving 5F - 9C = 160 For F: A Step-by-Step Guide

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In mathematics, solving for a variable in an equation is a fundamental skill. It allows us to isolate the variable and determine its value based on the given equation. This article focuses on providing a detailed, step-by-step solution to the equation 5F - 9C = 160 for F. This equation is a linear equation in two variables, F and C, and our goal is to express F in terms of C. By understanding the process of solving such equations, you'll be well-equipped to tackle various algebraic problems.

The process of solving linear equations involves applying algebraic operations to both sides of the equation while maintaining equality. The key is to isolate the variable you want to solve for on one side of the equation. This often involves combining like terms, using the distributive property, and performing inverse operations (addition, subtraction, multiplication, division). By following a systematic approach, you can confidently solve for any variable in a linear equation.

This article will meticulously walk you through the steps involved in isolating F in the equation 5F - 9C = 160. We will explain each step in detail, ensuring that you not only understand the solution but also grasp the underlying principles. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide you with the knowledge and confidence to solve similar equations. We'll break down the problem into manageable steps, making it easy to follow along and learn the techniques involved.

Understanding the Problem

Before diving into the solution, it's crucial to understand the problem statement. We are given the equation 5F - 9C = 160, which is a linear equation with two variables: F and C. Our objective is to solve this equation for F, meaning we want to isolate F on one side of the equation and express it in terms of C. This means we want to rewrite the equation in the form F = (some expression involving C).

Linear equations are equations where the highest power of any variable is 1. These equations can be represented graphically as a straight line. In our case, the equation 5F - 9C = 160 represents a linear relationship between F and C. Solving for F allows us to determine the value of F for any given value of C. This is particularly useful in various applications where relationships between variables need to be understood and quantified.

The given equation can be seen as a formula that relates two quantities, F and C. To solve for F, we need to use algebraic manipulations to undo the operations that are being performed on it. This involves using inverse operations and maintaining the balance of the equation. Remember, any operation performed on one side of the equation must also be performed on the other side to preserve equality. By carefully applying these principles, we can isolate F and find its expression in terms of C.

Step-by-Step Solution

Now, let's proceed with the step-by-step solution to solve for F in the equation 5F - 9C = 160.

  1. Isolate the term containing F:

    The first step is to isolate the term containing F, which is 5F. To do this, we need to eliminate the term -9C from the left side of the equation. We can achieve this by adding 9C to both sides of the equation. Adding the same quantity to both sides maintains the balance of the equation.

    • Original equation: 5F - 9C = 160
    • Add 9C to both sides: 5F - 9C + 9C = 160 + 9C
    • Simplify: 5F = 160 + 9C

    By adding 9C to both sides, we have successfully isolated the term 5F on the left side of the equation. This brings us one step closer to solving for F.

  2. Solve for F:

    Now that we have isolated the term 5F, we need to isolate F itself. The term 5F represents 5 times F. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 5.

    • Equation after step 1: 5F = 160 + 9C
    • Divide both sides by 5: (5F) / 5 = (160 + 9C) / 5
    • Simplify: F = (160 + 9C) / 5

    By dividing both sides by 5, we have isolated F on the left side of the equation. The right side now expresses F in terms of C.

  3. Simplify the expression (Optional):

    The expression (160 + 9C) / 5 can be further simplified by dividing each term in the numerator by 5.

    • Equation after step 2: F = (160 + 9C) / 5
    • Divide each term by 5: F = 160/5 + (9C)/5
    • Simplify: F = 32 + (9/5)C

    This simplified form expresses F as the sum of a constant term (32) and a term proportional to C ((9/5)C). While this simplification is optional, it can make the expression easier to work with in some contexts.

Final Answer

Therefore, the solution to the equation 5F - 9C = 160 for F is:

F = (160 + 9C) / 5

Or, in the simplified form:

F = 32 + (9/5)C

This is the final answer, expressing F in terms of C. This result allows us to calculate the value of F for any given value of C. The step-by-step solution provided here demonstrates the process of isolating a variable in a linear equation, a fundamental skill in algebra.

Key Takeaways

  • Solving for a variable involves isolating it on one side of the equation using algebraic operations.
  • Linear equations have variables raised to the power of 1 and can be represented as a straight line.
  • Inverse operations are crucial for isolating variables (e.g., addition and subtraction, multiplication and division).
  • Maintaining equality is essential; any operation performed on one side must be performed on the other.
  • Simplifying expressions can make them easier to work with and understand.

By mastering these concepts and techniques, you can confidently solve for any variable in linear equations and tackle more complex algebraic problems.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. Solve for x: 3x + 2y = 15
  2. Solve for p: 7p - 4q = 28
  3. Solve for m: 2m + 5n = -10

Working through these problems will help you apply the steps and techniques discussed in this article and further enhance your problem-solving skills. Remember to break down each problem into smaller steps, apply inverse operations, and maintain the balance of the equation.

Conclusion

Solving for a variable in an equation is a fundamental skill in algebra with wide-ranging applications. This article has provided a comprehensive guide to solving the equation 5F - 9C = 160 for F, illustrating the step-by-step process and underlying principles. By understanding these techniques, you can confidently tackle various algebraic problems and gain a deeper appreciation for the power of mathematical reasoning. Remember to practice regularly and apply these concepts to real-world scenarios to further enhance your skills.

This detailed guide should provide a solid foundation for understanding how to solve linear equations for a specific variable. With practice and a clear understanding of the underlying principles, you can confidently tackle a wide range of algebraic problems. Remember, mathematics is a skill that improves with practice, so keep working at it and you'll see your abilities grow.