Solving 11 - 4x = -9: A Step-by-Step Guide

by Alex Johnson 43 views

Let's dive into solving the equation 11 - 4x = -9. Math can sometimes seem daunting, but breaking it down step by step makes it much more manageable. This guide will walk you through the process, ensuring you understand each stage. We'll cover the basics of algebraic equations and then apply those principles to solve our specific problem. By the end, you'll not only have the answer but also a clearer understanding of how to tackle similar equations. So, grab a pen and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving our equation, it's essential to understand the basics of algebraic equations. An algebraic equation is a mathematical statement that shows the equality between two expressions. These expressions usually contain variables (like 'x' in our equation), constants (numbers), and mathematical operations (like addition, subtraction, multiplication, and division).

Key components of an algebraic equation include:

  • Variables: These are symbols (usually letters) that represent unknown values. In our equation, 'x' is the variable we want to find.
  • Constants: These are fixed numerical values. In 11 - 4x = -9, the constants are 11 and -9.
  • Coefficients: This is the number multiplied by the variable. In our equation, -4 is the coefficient of 'x'.
  • Operators: These are the symbols that indicate mathematical operations, such as +, -, ×, and ÷.

The goal of solving an algebraic equation is to isolate the variable on one side of the equation. This means getting the variable by itself so that we can determine its value. We achieve this by performing the same operations on both sides of the equation, maintaining the equality. Think of an equation like a balanced scale; whatever you do to one side, you must also do to the other to keep it balanced. This principle is the cornerstone of solving algebraic equations.

Understanding these fundamentals is crucial because it lays the groundwork for solving more complex equations in the future. Remember, the key is to keep the equation balanced and to systematically isolate the variable.

Step 1: Isolating the Term with the Variable

In our equation, 11 - 4x = -9, the first step is to isolate the term that contains the variable, which is -4x. To do this, we need to eliminate the constant term, which is 11, from the left side of the equation. We can achieve this by subtracting 11 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain balance.

So, we perform the operation:

11 - 4x - 11 = -9 - 11

This simplifies to:

-4x = -20

Now, we have successfully isolated the term with the variable. This is a crucial step because it brings us closer to finding the value of 'x'. By subtracting 11 from both sides, we've effectively moved the constant term to the right side of the equation, leaving only the term with 'x' on the left. This is a common strategy in solving algebraic equations: moving terms around to group like terms together.

Isolating the variable term is like clearing the path to our destination. It makes the next step, which is to isolate the variable itself, much more straightforward. Now that we have -4x = -20, we are just one step away from finding the value of 'x'. Remember, the key is to perform operations that simplify the equation while maintaining its balance. This foundational principle will help you solve a wide range of algebraic problems.

Step 2: Solving for the Variable 'x'

Now that we have the equation -4x = -20, the next step is to isolate 'x' completely. Currently, 'x' is being multiplied by -4. To undo this multiplication and get 'x' by itself, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -4.

So, we perform the operation:

(-4x) / -4 = -20 / -4

This simplifies to:

x = 5

And there you have it! We have successfully solved for 'x'. By dividing both sides of the equation by -4, we've isolated 'x' and found its value to be 5. This step highlights the importance of using inverse operations to unravel the equation. Multiplication and division are inverse operations, just like addition and subtraction. Using the correct inverse operation is crucial for isolating the variable and finding its value.

This step is often the final piece of the puzzle when solving linear equations. Once the variable is isolated, the solution is revealed. In this case, we've found that x = 5 is the solution to the equation 11 - 4x = -9. Remember, the goal is always to get the variable alone on one side of the equation. By applying the principles of inverse operations and maintaining balance, you can confidently solve for any variable in a linear equation.

Step 3: Verifying the Solution

After solving an equation, it's always a good practice to verify your solution. This ensures that you haven't made any mistakes along the way and that your answer is correct. To verify our solution, we'll substitute the value we found for 'x', which is 5, back into the original equation: 11 - 4x = -9.

Substitute x = 5:

11 - 4(5) = -9

Now, we simplify the left side of the equation:

11 - 20 = -9

-9 = -9

As you can see, the equation holds true. Both sides of the equation are equal, which confirms that our solution, x = 5, is correct. Verification is a crucial step in the problem-solving process. It provides you with the confidence that your answer is accurate and helps you catch any potential errors.

This step is particularly important in more complex equations where there are more opportunities for mistakes. By substituting your solution back into the original equation, you are essentially testing whether the value you found satisfies the equation's condition. If the equation remains balanced, your solution is correct. If not, you know there's an error somewhere in your steps, and you can go back and review your work. Always remember to verify your solutions to ensure accuracy and build your confidence in solving equations.

Conclusion: Mastering Algebraic Equations

We've successfully solved the equation 11 - 4x = -9, and the solution is x = 5. By following a step-by-step approach, we broke down the problem into manageable parts, making it easier to understand and solve. We started by understanding the basics of algebraic equations, then we isolated the term with the variable, solved for 'x', and finally, verified our solution.

Key takeaways from this process include:

  • Understanding the Basics: Knowing the components of an equation (variables, constants, coefficients, operators) is crucial.
  • Maintaining Balance: Remember to perform the same operations on both sides of the equation to keep it balanced.
  • Inverse Operations: Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable.
  • Verification: Always verify your solution by substituting it back into the original equation.

Solving algebraic equations is a fundamental skill in mathematics, and it's a skill that improves with practice. The more equations you solve, the more comfortable you'll become with the process. Don't be afraid to tackle challenging problems; each one is an opportunity to learn and grow. Remember, math is not just about finding the right answer; it's about understanding the process and developing problem-solving skills that you can apply in various areas of life.

By mastering algebraic equations, you're building a solid foundation for more advanced mathematical concepts. Keep practicing, stay curious, and you'll find that math can be both challenging and rewarding. If you want to learn more about algebra and equations, a great resource is Khan Academy's Algebra section.