Solving (2x - 15) / 4 = 12 A Step-by-Step Guide

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In the realm of mathematics, linear equations form a fundamental concept. These equations, characterized by a variable raised to the power of one, represent a straight line when graphed on a coordinate plane. Mastering the art of solving linear equations is crucial for success in algebra and beyond. This article will delve into the process of solving the specific linear equation (2x - 15) / 4 = 12, providing a comprehensive step-by-step guide suitable for learners of all levels. We will explore the underlying principles, demonstrate the solution process, and highlight key concepts to solidify your understanding. By the end of this guide, you will be well-equipped to tackle similar linear equations with confidence and accuracy.

Our focus is on solving the linear equation (2x - 15) / 4 = 12. This equation involves a single variable, x, and our goal is to isolate x on one side of the equation to determine its value. To achieve this, we will employ a series of algebraic operations, ensuring that we maintain the equality on both sides of the equation. Each step we take will bring us closer to the solution, revealing the value of x that satisfies the equation. Understanding the structure of the equation is the first step towards finding the solution. We have a fraction on the left side, and our initial strategies will involve eliminating this fraction to simplify the equation.

The presence of the fraction in the equation (2x - 15) / 4 = 12 complicates the process of isolating x. To eliminate this fraction, we employ a fundamental algebraic principle: multiplying both sides of the equation by the denominator. In this case, the denominator is 4. By multiplying both sides by 4, we effectively cancel out the division by 4 on the left side, simplifying the equation significantly. This step is crucial as it allows us to work with a more manageable equation, paving the way for further steps in solving for x. The equation now becomes:

4 * [(2x - 15) / 4] = 4 * 12

This simplifies to:

2x - 15 = 48

Now we have a simpler equation to work with, free from fractions.

Now that we have eliminated the fraction, the next step in solving for x is to isolate the term containing x. In the equation 2x - 15 = 48, the term containing x is 2x. To isolate this term, we need to eliminate the constant term, which is -15. We achieve this by performing the inverse operation: adding 15 to both sides of the equation. This maintains the equality and moves us closer to isolating x. This step is a key application of the addition property of equality. The equation becomes:

2x - 15 + 15 = 48 + 15

This simplifies to:

2x = 63

Now, the term with x is isolated on the left side, and we have a new constant on the right side.

We've reached the final step in solving for x. In the equation 2x = 63, x is being multiplied by 2. To isolate x completely, we need to perform the inverse operation: dividing both sides of the equation by 2. This utilizes the division property of equality and will give us the value of x that satisfies the original equation. This is the crucial step where we finally determine the solution to our linear equation. The equation becomes:

2x / 2 = 63 / 2

This simplifies to:

x = 31.5

Therefore, the solution to the equation (2x - 15) / 4 = 12 is x = 31.5.

To ensure the accuracy of our solution, it's crucial to verify it by substituting the value of x back into the original equation. This process confirms that our calculated value of x indeed satisfies the equation. We substitute x = 31.5 into the original equation (2x - 15) / 4 = 12:

(2 * 31.5 - 15) / 4 = 12

First, we calculate the expression inside the parentheses:

(63 - 15) / 4 = 12
48 / 4 = 12
12 = 12

The equation holds true, confirming that our solution x = 31.5 is correct. This verification step is a crucial practice in mathematics to ensure the accuracy of your solutions and build confidence in your problem-solving abilities.

In this guide, we've successfully solved the linear equation (2x - 15) / 4 = 12 using a step-by-step approach. We began by understanding the equation, then eliminated the fraction by multiplying both sides by 4. Next, we isolated the term with x by adding 15 to both sides. Finally, we solved for x by dividing both sides by 2, arriving at the solution x = 31.5. We then verified our solution by substituting it back into the original equation. This process illustrates the fundamental principles of solving linear equations, which involve performing inverse operations to isolate the variable. Mastering these principles will empower you to tackle a wide range of mathematical problems with confidence. Remember, practice is key to developing your skills in solving linear equations and other algebraic concepts. Continue to work through various examples and challenge yourself with increasingly complex problems to strengthen your understanding and problem-solving abilities. Keep practicing and exploring the world of mathematics!