Introduction
In the realm of mathematics, linear equations form the bedrock of many algebraic concepts. Understanding how to solve these equations is crucial for anyone venturing further into math, science, or engineering. This article delves into the process of solving a specific linear equation: 3(-4x + 5) = 12. We will break down each step, providing a clear and comprehensive guide that will equip you with the skills to tackle similar problems with confidence. Our focus will be on delivering a high-quality, value-driven explanation suitable for learners of all levels. We'll start by defining what a linear equation is, then methodically solve the given equation, and finally, discuss the importance of verifying our solution. So, if you're ready to enhance your problem-solving capabilities, let's embark on this mathematical journey together! Remember, the key to mastering mathematics lies in consistent practice and a thorough understanding of fundamental principles. This guide aims to provide you with both, ensuring you not only learn the steps but also understand the reasoning behind them. Through careful explanations and clear examples, we aim to demystify the process of solving linear equations and empower you to approach mathematical challenges with greater assurance.
Understanding Linear Equations
Before we dive into solving the equation 3(-4x + 5) = 12, it’s crucial to grasp the concept of linear equations. A linear equation is essentially an algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations are termed "linear" because they describe a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to find. The highest power of the variable in a linear equation is always 1. This distinguishes them from quadratic equations (where the highest power is 2), cubic equations (where the highest power is 3), and so on. The beauty of linear equations lies in their simplicity and predictability. They have a straightforward solution, and the methods for solving them are well-established. Mastering these techniques is a fundamental step in algebra. To solve a linear equation, our primary goal is to isolate the variable on one side of the equation. This means manipulating the equation using algebraic operations until we have 'x' (or whatever the variable is) alone on one side, with its value on the other side. This process typically involves using inverse operations to undo the operations performed on the variable. For instance, if the variable is being multiplied by a number, we divide both sides of the equation by that number. Understanding this core principle is essential for effectively tackling any linear equation. In the subsequent sections, we will apply this principle to solve the specific equation presented, illustrating each step in detail.
Solving 3(-4x + 5) = 12: A Step-by-Step Approach
Now, let's tackle the equation 3(-4x + 5) = 12 head-on. We'll break down the solution into manageable steps, ensuring clarity and comprehension at every stage. First, we need to address the parentheses. The equation includes a term in parentheses multiplied by a constant, so our initial step is to distribute the 3 across the terms inside the parentheses. This means multiplying both -4x and 5 by 3. This gives us 3 * -4x + 3 * 5 = 12, which simplifies to -12x + 15 = 12. Distributing correctly is crucial for maintaining the balance of the equation. Next, our aim is to isolate the term containing 'x'. Currently, we have -12x + 15 = 12. To get the -12x term alone, we need to eliminate the +15 on the left side. We can do this by subtracting 15 from both sides of the equation. This ensures we maintain the equality. Subtracting 15 from both sides gives us -12x + 15 - 15 = 12 - 15, which simplifies to -12x = -3. Now, we have the term with 'x' isolated, but 'x' is still being multiplied by -12. To completely isolate 'x', we need to undo this multiplication. We achieve this by dividing both sides of the equation by -12. Dividing both sides by -12 gives us -12x / -12 = -3 / -12. This simplifies to x = 1/4. Therefore, the solution to the equation 3(-4x + 5) = 12 is x = 1/4. This step-by-step approach highlights the systematic process involved in solving linear equations. In the next section, we'll emphasize the vital practice of verifying our solution to ensure accuracy.
Verifying the Solution
After finding a solution to an equation, it's essential to verify its correctness. This step is a crucial safeguard against errors and ensures that the value we've calculated for 'x' truly satisfies the original equation. To verify our solution, we substitute the value we found (x = 1/4) back into the original equation, which is 3(-4x + 5) = 12. Substituting x = 1/4, we get 3(-4 * (1/4) + 5) = 12. Now, we simplify the expression inside the parentheses first. -4 * (1/4) equals -1. So, the equation becomes 3(-1 + 5) = 12. Next, we simplify the parentheses further: -1 + 5 equals 4. Our equation now looks like this: 3 * 4 = 12. Multiplying 3 by 4 gives us 12, so we have 12 = 12. This final statement is true, indicating that our solution, x = 1/4, is indeed correct. The left side of the equation equals the right side when we substitute our value for 'x'. This verification process not only confirms the accuracy of our solution but also reinforces our understanding of the equation and its properties. It's a practice that should be adopted as a standard part of problem-solving in mathematics. By verifying our solutions, we develop confidence in our abilities and minimize the risk of carrying errors forward in more complex calculations. In the following section, we will summarize the key steps and discuss the broader implications of understanding linear equations.
Conclusion: The Power of Linear Equations
In summary, we've successfully navigated the process of solving the linear equation 3(-4x + 5) = 12. We began by distributing the 3 across the terms inside the parentheses, resulting in -12x + 15 = 12. We then isolated the term with 'x' by subtracting 15 from both sides, which gave us -12x = -3. Finally, we solved for 'x' by dividing both sides by -12, leading to the solution x = 1/4. Crucially, we verified our solution by substituting x = 1/4 back into the original equation and confirming that it held true. This step-by-step approach underscores the systematic nature of solving linear equations. Each operation is performed with the aim of isolating the variable, and maintaining the balance of the equation is paramount throughout the process. Understanding and mastering linear equations is a foundational skill in mathematics. These equations form the basis for more advanced concepts in algebra and calculus, and they have wide-ranging applications in various fields, including physics, engineering, economics, and computer science. The ability to confidently solve linear equations opens doors to tackling more complex mathematical problems and real-world challenges. Furthermore, the problem-solving skills honed through working with linear equations – such as breaking down a problem into smaller steps, applying logical operations, and verifying results – are transferable to many other areas of life. Therefore, the time and effort invested in understanding linear equations are well worth it. They empower us to approach mathematical problems with clarity and precision, and they provide a solid foundation for future learning. As you continue your mathematical journey, remember the principles we've discussed here, and practice consistently to strengthen your skills. The world of mathematics is vast and fascinating, and a strong understanding of fundamentals like linear equations is your key to unlocking its many wonders.