Solving 3x - 5 + 2x = 25 A Step-by-Step Guide

Linear equations are fundamental to mathematics and form the basis for many real-world applications. The ability to solve these equations is a crucial skill for students and professionals alike. In this article, we will delve into the process of solving the linear equation $3x - 5 + 2x = 25$, providing a step-by-step guide to arrive at the correct solution. We will not only determine the value of $x$ that satisfies the equation but also discuss the underlying principles and techniques involved in solving such problems. Understanding these principles is essential for tackling more complex equations and mathematical challenges in the future.

Before we dive into the specific equation, it’s important to understand the basic concepts. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is typically represented by letters such as $x$, $y$, or $z$. Solving a linear equation involves finding the value of the variable that makes the equation true. This is achieved by isolating the variable on one side of the equation through a series of algebraic manipulations. These manipulations must maintain the equality of both sides of the equation, ensuring that the solution remains valid. The key operations include combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by the same non-zero value. Each of these steps is designed to simplify the equation and bring us closer to isolating the variable.

Furthermore, the process of solving linear equations reinforces fundamental algebraic principles such as the commutative, associative, and distributive properties. These properties allow us to rearrange and simplify terms within the equation. For instance, the commutative property allows us to change the order of terms (e.g., $a + b = b + a$), while the associative property allows us to regroup terms (e.g., $(a + b) + c = a + (b + c)$). The distributive property is crucial for dealing with expressions involving parentheses (e.g., $a(b + c) = ab + ac$). Mastering these properties is essential for manipulating equations and solving for the unknown variable effectively. By understanding and applying these concepts, you will be well-equipped to solve a wide range of linear equations and related mathematical problems.

Solving the equation $3x - 5 + 2x = 25$ requires a systematic approach. Our primary goal is to isolate the variable $x$ on one side of the equation. To achieve this, we will follow a series of algebraic steps, ensuring that each operation maintains the equality of the equation. By carefully executing these steps, we can accurately determine the value of $x$ that satisfies the equation. This methodical approach not only helps in solving the current problem but also builds a strong foundation for tackling more complex equations in the future. Each step is designed to simplify the equation, bringing us closer to the solution while reinforcing essential algebraic principles.

The first step in solving this equation involves combining like terms. Like terms are those that contain the same variable raised to the same power. In this equation, $3x$ and $2x$ are like terms. By combining them, we simplify the equation and make it easier to work with. This process involves adding the coefficients of the like terms, which in this case are 3 and 2. The sum of these coefficients is 5, so we can rewrite the equation as $5x - 5 = 25$. This step is crucial as it reduces the number of terms in the equation, making subsequent steps more straightforward. Combining like terms is a fundamental technique in algebra and is used extensively in solving various types of equations. It allows us to consolidate the equation and focus on the essential components needed to isolate the variable.

Following the combination of like terms, the next step is to isolate the term containing $x$. In our simplified equation, $5x - 5 = 25$, we need to eliminate the constant term (-5) from the left side. To do this, we apply the addition property of equality, which states that adding the same value to both sides of an equation maintains the equality. In this case, we add 5 to both sides of the equation. This results in $5x - 5 + 5 = 25 + 5$, which simplifies to $5x = 30$. This step is crucial because it brings us closer to isolating the variable $x$. By adding 5 to both sides, we effectively cancel out the -5 on the left side, leaving us with a term that only involves $x$. This manipulation is a key technique in solving linear equations and is based on the principle of maintaining balance on both sides of the equation.

Finally, to completely isolate $x$, we need to get rid of the coefficient 5. To do this, we apply the division property of equality, which states that dividing both sides of an equation by the same non-zero value maintains the equality. In this case, we divide both sides of the equation $5x = 30$ by 5. This gives us rac{5x}{5} = rac{30}{5}, which simplifies to $x = 6$. This is the final step in solving the equation, and it reveals the value of $x$ that satisfies the original equation. Dividing both sides by the coefficient of $x$ is a standard procedure in solving linear equations and ensures that we obtain the correct solution. The result, $x = 6$, is the value that makes the original equation true, and it can be verified by substituting it back into the equation.

Verifying the solution is a crucial step in solving any equation. It ensures that the value we have found for the variable actually satisfies the original equation. This process helps to identify any potential errors made during the solving process and provides confidence in the accuracy of our solution. In the case of the equation $3x - 5 + 2x = 25$, we found that $x = 6$. To verify this solution, we will substitute 6 for $x$ in the original equation and check if both sides of the equation are equal. This step is not just a formality but an integral part of the problem-solving process, reinforcing the understanding of how solutions work and highlighting the importance of precision in algebraic manipulations.

To verify the solution $x = 6$, we substitute 6 for $x$ in the original equation $3x - 5 + 2x = 25$. This substitution gives us $3(6) - 5 + 2(6) = 25$. Now, we perform the operations on the left side of the equation following the order of operations (PEMDAS/BODMAS). First, we perform the multiplications: $3(6) = 18$ and $2(6) = 12$. So, the equation becomes $18 - 5 + 12 = 25$. Next, we perform the addition and subtraction from left to right. Subtracting 5 from 18 gives us 13, so the equation is now $13 + 12 = 25$. Finally, adding 13 and 12 gives us 25, so the left side of the equation simplifies to 25. This matches the right side of the equation, which is also 25.

Since substituting $x = 6$ into the original equation results in both sides being equal (25 = 25), we can confidently conclude that our solution is correct. This verification process confirms that 6 is indeed the value of $x$ that satisfies the equation $3x - 5 + 2x = 25$. This step-by-step substitution and simplification not only validates our solution but also reinforces our understanding of the equation's structure and the algebraic principles involved. By verifying the solution, we ensure that we have not made any mistakes in our calculations and that our final answer is accurate. This practice is essential for developing strong problem-solving skills in mathematics and related fields.

Identifying the correct answer choice from the given options is the final step in solving the problem. After determining the solution to the equation, it is essential to compare our result with the provided options to select the one that matches. This step not only confirms our solution but also tests our ability to accurately interpret and apply the answer choices within the context of the problem. In this case, we have found that $x = 6$ is the solution to the equation $3x - 5 + 2x = 25$. We now need to check which of the given answer choices corresponds to this value. This process reinforces the importance of accuracy and attention to detail in mathematical problem-solving.

The answer choices provided are:

A. $x = 6$ B. $x = 30$ C. $x = 4$ D. $x = 20$

By comparing our solution, $x = 6$, with the given options, we can clearly see that option A, $x = 6$, is the correct answer. The other options, $x = 30$, $x = 4$, and $x = 20$, do not match our solution and would not satisfy the original equation if substituted. Therefore, selecting option A as the correct answer is the logical and accurate choice. This step highlights the importance of not only finding the correct solution but also accurately identifying it within the context of the provided answer choices. It reinforces the need for careful comparison and attention to detail in mathematical problem-solving, ensuring that we select the correct option based on our calculations and understanding of the equation.

In conclusion, solving the equation $3x - 5 + 2x = 25$ has provided a valuable opportunity to reinforce fundamental algebraic skills. Through a step-by-step approach, we successfully determined that $x = 6$ is the solution. This process involved combining like terms, isolating the variable, and verifying the solution, all of which are essential techniques in algebra. The ability to solve linear equations is not only crucial for academic success but also for various real-world applications. From everyday problem-solving to advanced scientific calculations, algebraic skills form the foundation for critical thinking and analytical reasoning. This exercise underscores the importance of mastering these skills to effectively tackle mathematical challenges and develop a deeper understanding of the subject.

The key takeaways from this exercise include the importance of a systematic approach to solving equations, the significance of combining like terms, the necessity of isolating the variable, and the value of verifying the solution. Each step in the process is designed to simplify the equation and bring us closer to the correct answer. Combining like terms allows us to consolidate the equation, making it easier to manipulate. Isolating the variable is the core objective of solving any equation, and it involves performing inverse operations to remove other terms from the same side of the equation. Verifying the solution ensures that our answer is accurate and satisfies the original equation. These principles are applicable to a wide range of algebraic problems and are essential for developing proficiency in mathematics.

Furthermore, the ability to solve linear equations is a gateway to more advanced mathematical concepts. As we progress in our mathematical journey, we will encounter more complex equations and systems of equations that build upon these fundamental skills. Understanding how to manipulate and solve linear equations is a prerequisite for tackling quadratic equations, inequalities, and more advanced algebraic topics. Moreover, these skills are crucial in fields such as physics, engineering, economics, and computer science, where mathematical modeling and problem-solving are essential. Therefore, investing time and effort in mastering algebraic skills is a worthwhile endeavor that will pay dividends in both academic and professional pursuits. The systematic approach and logical reasoning developed through solving equations are valuable assets that can be applied to a wide range of problem-solving scenarios.