Solving (2/5)x - 17 = 15 Step-by-Step A Comprehensive Guide
In mathematics, solving for an unknown variable is a fundamental skill. This article delves into a specific problem: finding the value of x in the linear equation (2/5) x - 17 = 15. We will break down the problem step-by-step, explaining the underlying principles and providing a clear, concise solution. This approach is crucial not only for solving this particular equation but also for understanding the broader concepts of algebra.
Understanding Linear Equations
Before diving into the solution, let's first define what a linear equation is. 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. In simpler terms, it's an equation that can be graphed as a straight line. The equation (2/5) x - 17 = 15 fits this definition, as it involves a single variable (x) raised to the power of 1.
Linear equations are pervasive in various fields, from physics and engineering to economics and computer science. They are used to model relationships between quantities that change at a constant rate. Therefore, mastering the techniques for solving these equations is essential for anyone pursuing studies or careers in these areas. Understanding the properties of linear equations allows us to manipulate them algebraically to isolate the variable we are trying to solve for. This process often involves performing the same operations on both sides of the equation to maintain equality. The goal is to simplify the equation until the variable is on one side and a constant value is on the other.
The key to solving linear equations lies in the principle of inverse operations. Each mathematical operation has an inverse that undoes it. For example, addition and subtraction are inverse operations, as are multiplication and division. By applying inverse operations in the correct order, we can systematically isolate the variable. In the given equation, we will use addition to eliminate the constant term and multiplication to eliminate the fractional coefficient.
The Problem: (2/5)x - 17 = 15
Our goal is to isolate x on one side of the equation. To do this, we will use the principles of inverse operations. The first step involves addressing the constant term, -17. To eliminate it, we add 17 to both sides of the equation.
Step 1: Isolating the Term with x
To begin solving for x in the equation (2/5) x - 17 = 15, our initial focus is on isolating the term containing x. This means we need to eliminate the constant term, -17, which is being subtracted from (2/5) x. The inverse operation of subtraction is addition, so we will add 17 to both sides of the equation. This is a critical step in solving linear equations, as it maintains the balance of the equation while moving us closer to isolating the variable.
Adding 17 to both sides ensures that the equation remains balanced. Remember, any operation performed on one side of the equation must also be performed on the other side to maintain equality. This principle is fundamental to algebraic manipulation. When we add 17 to the left side, -17 cancels out, leaving us with (2/5) x. On the right side, 15 + 17 equals 32. Therefore, the equation now becomes (2/5) x = 32. This simplified equation represents a significant step forward in our solution. We have successfully isolated the term containing x, setting the stage for the next step: eliminating the fractional coefficient.
This step highlights the importance of understanding inverse operations. By recognizing that addition is the inverse of subtraction, we were able to strategically eliminate the constant term. This approach is applicable to a wide range of linear equations, making it a valuable tool in your mathematical arsenal. The ability to manipulate equations by applying inverse operations is a cornerstone of algebra and a skill that will serve you well in more advanced mathematical concepts.
Step 2: Eliminating the Fractional Coefficient
Now we have the equation (2/5) x = 32. Our next objective is to eliminate the fractional coefficient (2/5) that is multiplying x. To do this, we need to perform the inverse operation of multiplication, which is division. However, instead of dividing by a fraction, it's often more convenient to multiply by the reciprocal of the fraction. The reciprocal of (2/5) is (5/2). This strategy is particularly useful when dealing with linear equations involving fractions.
Multiplying both sides of the equation by (5/2) will effectively cancel out the fractional coefficient on the left side. When we multiply (2/5) x by (5/2), the 2 in the numerator and denominator cancel out, as do the 5s. This leaves us with just x on the left side, which is exactly what we want. On the right side, we multiply 32 by (5/2). This can be done by first dividing 32 by 2, which gives us 16, and then multiplying 16 by 5, resulting in 80. Therefore, the equation simplifies to x = 80. This solution represents the value of x that satisfies the original equation.
This step demonstrates the power of reciprocals in simplifying equations. By understanding the relationship between a fraction and its reciprocal, we can efficiently eliminate fractional coefficients. This technique is not only applicable to linear equations but also to more complex algebraic expressions. The concept of reciprocals is a fundamental building block in algebra and is essential for solving a wide variety of mathematical problems. The ability to confidently manipulate fractions and reciprocals is a key skill for success in mathematics.
The Solution: x = 80
After performing these steps, we arrive at the solution: x = 80. This means that the value of x that satisfies the equation (2/5) x - 17 = 15 is 80. To ensure the accuracy of our solution, it's always a good practice to check our answer by substituting it back into the original equation. This verification step provides a crucial confirmation that our algebraic manipulations were correct and that we have indeed found the correct value of x.
Step 3: Verifying the Solution
To verify our solution, we substitute x = 80 back into the original equation: (2/5) x - 17 = 15. Replacing x with 80, we get (2/5) * 80 - 17. First, we multiply (2/5) by 80. This can be done by dividing 80 by 5, which gives us 16, and then multiplying 16 by 2, resulting in 32. So, the equation becomes 32 - 17. Subtracting 17 from 32, we get 15. This confirms that our left side of the equation equals the right side (15 = 15), indicating that our solution x = 80 is correct. This verification process is a crucial step in solving linear equations and other algebraic problems.
Checking our solution not only confirms its accuracy but also reinforces our understanding of the equation and the steps we took to solve it. It helps to solidify the concepts and techniques involved in algebraic manipulation. By substituting the solution back into the original equation, we are essentially reversing the steps we took to isolate x. If the equation holds true, it provides strong evidence that our solution is correct. This practice is highly recommended for all mathematical problems, as it can help prevent errors and build confidence in your problem-solving abilities.
In addition to verifying the solution, it's also beneficial to reflect on the steps taken to solve the equation. This metacognitive process helps to identify the key concepts and strategies involved, such as inverse operations and reciprocals. By understanding the underlying principles, you can apply these techniques to solve other linear equations and more complex mathematical problems. The ability to solve linear equations is a fundamental skill in algebra, and mastering this skill will pave the way for success in higher-level mathematics.
Conclusion
In conclusion, we have successfully solved the equation (2/5) x - 17 = 15 and found that x = 80. We accomplished this by applying the principles of inverse operations, first adding 17 to both sides to isolate the term with x, and then multiplying both sides by the reciprocal of the fractional coefficient to solve for x. Finally, we verified our solution by substituting it back into the original equation. This step-by-step approach provides a clear and concise method for solving linear equations and highlights the importance of understanding the underlying mathematical principles.
This problem demonstrates the power of algebraic manipulation and the importance of following a systematic approach. By breaking down the problem into smaller, manageable steps, we were able to isolate the variable and find its value. The techniques used in this example can be applied to a wide range of linear equations, making them valuable tools for problem-solving in mathematics and other fields. The ability to confidently solve linear equations is a fundamental skill that will serve you well in your mathematical journey.
The process of solving this equation also reinforces the importance of checking your work. The verification step is crucial for ensuring the accuracy of your solution and for identifying any potential errors. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and that our algebraic manipulations were correct. This practice not only builds confidence in your problem-solving abilities but also helps to develop a deeper understanding of the concepts involved. Remember, mathematics is not just about finding the right answer; it's also about understanding the process and being able to explain your reasoning. By following a systematic approach and verifying your solutions, you can develop strong mathematical skills and achieve success in your studies.
