Solving 4(5x + 3) = 14x + 30 A Step By Step Guide

What is the solution to the equation 4(5x + 3) = 14x + 30?

Introduction: Unveiling the Solution to the Equation

In this comprehensive guide, we delve into the step-by-step process of solving the equation 4(5x + 3) = 14x + 30. This equation, a fundamental concept in algebra, involves variables, coefficients, and constants, and finding its solution requires a systematic approach. The solution to an equation is the value or values of the variable that make the equation true. In this case, we are looking for the value of 'x' that satisfies the given equation. Understanding how to solve such equations is crucial for various mathematical applications and problem-solving scenarios. This article aims to provide a clear and detailed explanation of the solution process, making it accessible to learners of all levels. We will break down each step, ensuring that you not only understand the solution but also the underlying principles of algebraic manipulation.

Our journey will begin by distributing the constant on the left side of the equation, a crucial step in simplifying the expression. We will then combine like terms, isolating the variable 'x' on one side of the equation. This process involves carefully applying the properties of equality, ensuring that we maintain the balance of the equation throughout. Finally, we will arrive at the solution for 'x', which will be one of the options provided. We will also discuss how to verify the solution, a vital step in ensuring the accuracy of our answer. By the end of this guide, you will have a solid understanding of how to solve this specific equation and similar algebraic problems.

Furthermore, we will explore the broader context of solving linear equations, highlighting the importance of these skills in various fields, from science and engineering to economics and finance. The ability to manipulate equations and solve for unknowns is a fundamental tool in quantitative reasoning and problem-solving. This guide will not only provide you with the answer to this specific question but also equip you with the skills and knowledge to tackle a wide range of algebraic challenges. So, let's embark on this mathematical journey and unravel the solution to the equation 4(5x + 3) = 14x + 30.

Step-by-Step Solution: Deciphering the Algebraic Puzzle

To effectively solve the equation 4(5x + 3) = 14x + 30, we must embark on a step-by-step journey, meticulously applying the rules of algebra to isolate the variable 'x'. This process involves several key stages, each building upon the previous one, to ultimately reveal the solution. Let's begin by addressing the parentheses on the left side of the equation. This initial step, known as distribution, is crucial for simplifying the expression and paving the way for further manipulation. By distributing the constant 4 across the terms within the parentheses, we effectively eliminate the grouping and transform the equation into a more manageable form.

The first step in our solution is to distribute the 4 on the left side of the equation. This involves multiplying 4 by both terms inside the parentheses: 5x and 3. Performing this operation, we get: 4 * 5x = 20x and 4 * 3 = 12. Thus, the left side of the equation becomes 20x + 12. Now, our equation looks like this: 20x + 12 = 14x + 30. This transformation is a crucial step in simplifying the equation and bringing us closer to isolating the variable 'x'.

Next, we need to collect the 'x' terms on one side of the equation and the constant terms on the other. This is achieved by adding or subtracting terms from both sides of the equation, maintaining the balance and equality. To move the 'x' terms to the left side, we subtract 14x from both sides of the equation. This gives us: 20x - 14x + 12 = 14x - 14x + 30, which simplifies to 6x + 12 = 30. Now, we have all the 'x' terms on the left side and are ready to isolate 'x' further.

Following this, we isolate the 'x' term by moving the constant term, 12, to the right side of the equation. We do this by subtracting 12 from both sides: 6x + 12 - 12 = 30 - 12. This simplifies to 6x = 18. We are now just one step away from finding the value of 'x'. Finally, to solve for 'x', we divide both sides of the equation by the coefficient of 'x', which is 6. This gives us: 6x / 6 = 18 / 6, which simplifies to x = 3. Therefore, the solution to the equation 4(5x + 3) = 14x + 30 is x = 3. This step-by-step approach ensures that we arrive at the correct solution while maintaining a clear understanding of the algebraic principles involved.

Verification: Ensuring the Accuracy of Our Solution

After solving an equation, it is always prudent to verify the solution. This crucial step ensures that the value we have found for the variable 'x' indeed satisfies the original equation. Verification involves substituting the solution back into the equation and checking if both sides of the equation are equal. This process not only confirms the accuracy of our calculations but also reinforces our understanding of the equation and its solution. In the case of the equation 4(5x + 3) = 14x + 30, we found that x = 3. To verify this solution, we will substitute x = 3 back into the original equation and evaluate both sides.

Let's begin by substituting x = 3 into the left side of the equation: 4(5x + 3). Replacing 'x' with 3, we get 4(5 * 3 + 3). Following the order of operations, we first evaluate the expression inside the parentheses: 5 * 3 = 15, so we have 4(15 + 3). Next, we add 15 and 3, which gives us 18. So, the left side becomes 4 * 18. Multiplying 4 by 18, we get 72. Thus, the left side of the equation evaluates to 72 when x = 3.

Now, let's substitute x = 3 into the right side of the equation: 14x + 30. Replacing 'x' with 3, we get 14 * 3 + 30. First, we multiply 14 by 3, which gives us 42. So, the right side becomes 42 + 30. Adding 42 and 30, we get 72. Therefore, the right side of the equation also evaluates to 72 when x = 3.

Since both the left side and the right side of the equation equal 72 when x = 3, we can confidently conclude that our solution is correct. This verification process provides us with assurance that our algebraic manipulations were accurate and that x = 3 is indeed the solution to the equation 4(5x + 3) = 14x + 30. This practice of verification is an essential habit in mathematics, ensuring the reliability of our results and enhancing our problem-solving skills.

Conclusion: Embracing the Power of Algebraic Solutions

In conclusion, we have successfully navigated the process of solving the equation 4(5x + 3) = 14x + 30, arriving at the solution x = 3. This journey has taken us through the essential steps of algebraic manipulation, from distributing constants and combining like terms to isolating the variable and verifying our solution. The ability to solve such equations is a cornerstone of mathematical proficiency, with applications extending far beyond the classroom. From scientific research and engineering design to financial analysis and economic forecasting, the principles of algebra underpin a vast array of disciplines.

The key to solving equations like this lies in a systematic approach. By following a clear, step-by-step process, we can break down complex problems into manageable components. Distribution, combining like terms, and isolating the variable are fundamental techniques that can be applied to a wide range of algebraic equations. Furthermore, the importance of verification cannot be overstated. By substituting our solution back into the original equation, we ensure the accuracy of our work and gain confidence in our problem-solving abilities.

This exercise also highlights the power of algebra as a tool for problem-solving. Equations are mathematical statements that express relationships between variables and constants. By manipulating these equations, we can uncover hidden information and solve for unknown quantities. The skills we have developed in this process are not only valuable in mathematics but also in critical thinking and logical reasoning. As we continue our mathematical journey, we will encounter increasingly complex equations and problems. However, the fundamental principles and techniques we have learned here will serve as a solid foundation for future success.

Therefore, the solution to the equation 4(5x + 3) = 14x + 30 is indeed x = 3, option D. This exercise has not only provided us with an answer but also reinforced our understanding of algebraic problem-solving and the importance of a systematic and rigorous approach. Embracing these skills will empower us to tackle a wide range of mathematical challenges and unlock the power of algebraic solutions in various contexts.