Solving 9 - 4x = 57 Step-by-Step Guide

by ADMIN 39 views

Finding the value of x in the equation 9 - 4x = 57 is a fundamental algebraic problem that demonstrates core principles of equation solving. This article will delve into a step-by-step approach to solving this equation, ensuring clarity and understanding for anyone encountering similar problems. We'll break down each step, explaining the logic behind it, and highlighting common pitfalls to avoid. Mastering this type of equation is crucial for building a strong foundation in algebra and more advanced mathematics.

Understanding the Basics of Algebraic Equations

Before diving into the specific solution, let's recap some essential concepts about algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. The goal of solving an equation is to isolate the variable (in this case, x) on one side of the equation to determine its value. To achieve this, we use inverse operations, which are operations that undo each other. The key principle is to maintain the balance of the equation; any operation performed on one side must also be performed on the other side to preserve the equality.

The Properties of Equality

Understanding the properties of equality is fundamental to solving algebraic equations. These properties allow us to manipulate equations without changing their solutions. There are four main properties of equality:

  1. Addition Property of Equality: If a = b, then a + c = b + c for any real number c. This means you can add the same value to both sides of an equation.
  2. Subtraction Property of Equality: If a = b, then a - c = b - c for any real number c. This means you can subtract the same value from both sides of an equation.
  3. Multiplication Property of Equality: If a = b, then ac = bc for any real number c. This means you can multiply both sides of an equation by the same value.
  4. Division Property of Equality: If a = b, then a / c = b / c for any non-zero real number c. This means you can divide both sides of an equation by the same non-zero value.

These properties are the tools we'll use to isolate x in the given equation. Now, let's proceed with solving the equation step-by-step.

Step-by-Step Solution of the Equation 9 - 4x = 57

Our equation is 9 - 4x = 57. Our goal is to isolate x on one side of the equation. Here's how we'll do it:

Step 1: Isolate the Term with x

The first step is to isolate the term containing x, which is -4x. To do this, we need to eliminate the constant term, which is 9, from the left side of the equation. Since 9 is being added (or subtracted, depending on how you look at it), we'll use the subtraction property of equality. We subtract 9 from both sides of the equation:

9 - 4x - 9 = 57 - 9

This simplifies to:

-4x = 48

Step 2: Solve for x

Now that we have isolated the term with x, we need to solve for x itself. The coefficient of x is -4, which means x is being multiplied by -4. To undo this multiplication, we use the division property of equality. We divide both sides of the equation by -4:

-4x / -4 = 48 / -4

This simplifies to:

x = -12

Therefore, the value of x that satisfies the equation 9 - 4x = 57 is -12. We can verify this solution by substituting -12 back into the original equation.

Verification of the Solution

To verify our solution, we substitute x = -12 back into the original equation 9 - 4x = 57:

9 - 4(-12) = 57

Now, we simplify the left side of the equation:

9 + 48 = 57

57 = 57

Since the left side equals the right side, our solution x = -12 is correct. This verification step is crucial because it ensures that we haven't made any errors in our calculations.

Common Mistakes to Avoid

When solving algebraic equations, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

  1. Incorrectly Applying the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Make sure you perform multiplication and division before addition and subtraction.
  2. Forgetting to Apply Operations to Both Sides: Remember, any operation you perform on one side of the equation must also be performed on the other side to maintain equality.
  3. Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. A simple sign error can lead to an incorrect solution.
  4. Dividing by Zero: Never divide by zero. It's undefined and will lead to mathematical errors.
  5. Skipping Steps: While it may be tempting to skip steps to save time, it's often better to write out each step to minimize errors.

By being aware of these common mistakes, you can increase your accuracy and confidence in solving algebraic equations.

Alternative Approaches to Solving the Equation

While the step-by-step method outlined above is a standard approach, there are alternative ways to think about solving the equation 9 - 4x = 57. One such approach involves rearranging the equation slightly before starting the isolation process. Let's explore this alternative:

  1. Rearranging the Equation: Instead of directly subtracting 9 from both sides, we can add 4x to both sides to get the x term on the right side:

    9 - 4x + 4x = 57 + 4x

    This simplifies to:

    9 = 57 + 4x

  2. Isolating the Term with x: Now, we subtract 57 from both sides to isolate the 4x term:

    9 - 57 = 57 + 4x - 57

    This simplifies to:

    -48 = 4x

  3. Solving for x: Finally, we divide both sides by 4 to solve for x:

    -48 / 4 = 4x / 4

    This simplifies to:

    x = -12

This alternative approach arrives at the same solution, x = -12, but it demonstrates that there can be multiple valid paths to solving an equation. Choosing the most efficient method often depends on personal preference and the specific structure of the equation.

The Importance of Practice and Conceptual Understanding

Solving algebraic equations, like 9 - 4x = 57, requires both procedural skill and conceptual understanding. While memorizing steps can be helpful, it's equally important to understand why those steps work. This understanding allows you to adapt your approach to different types of equations and problems.

Practice is key to mastering algebraic equation solving. The more you practice, the more comfortable you'll become with the different techniques and strategies. Work through various examples, and don't be afraid to make mistakes – they're valuable learning opportunities.

Conceptual understanding involves grasping the underlying principles of algebra, such as the properties of equality and the order of operations. When you understand these concepts, you can approach problems with confidence and flexibility.

Conclusion: Mastering Algebraic Equations

Solving the equation 9 - 4x = 57 is a prime example of a fundamental algebraic skill. By following a step-by-step approach, understanding the properties of equality, and avoiding common mistakes, you can confidently find the value of x. Remember to verify your solution to ensure accuracy and to practice regularly to build your skills.

Algebraic equations are the building blocks of more advanced mathematics, so mastering them is crucial for your mathematical journey. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this comprehensive guide provides the tools and knowledge you need to succeed. The solution to the equation 9 - 4x = 57 is x = -12, which corresponds to option C.