Solve The Equation -10x + 5 = 15 For X.

In this article, we will walk through the steps to solve for x in the equation f(x) = -10x + 5, given that f(x) = 15. This is a common problem in algebra, and understanding how to solve it is crucial for mastering more complex mathematical concepts. Our main objective is to isolate x on one side of the equation, which will reveal its value. This process involves substituting the given value of f(x), simplifying the equation, and performing basic algebraic manipulations such as addition, subtraction, multiplication, and division. This step-by-step guide will make the solution clear and easy to follow, even if you're new to algebra. By breaking down the problem into manageable steps, we aim to provide a comprehensive understanding of how to tackle similar equations in the future. This problem is a fundamental example of how functions and algebraic equations are used in mathematics, making it an essential topic for students and anyone interested in enhancing their problem-solving skills. We will also discuss common mistakes and provide tips to avoid them, ensuring a solid grasp of the concepts involved. Furthermore, we'll highlight the significance of this type of problem in real-world applications, demonstrating how algebraic skills are relevant beyond the classroom. Understanding the process of solving for x in linear equations is not just an academic exercise; it's a practical skill that can be applied in various fields, from finance to engineering. This article will equip you with the knowledge and confidence to solve similar problems and appreciate the broader applications of algebra.

1. Substitute the Value of f(x)

We are given the function f(x) = -10x + 5 and the condition f(x) = 15. The first step in solving for x is to substitute the value of f(x) into the equation. This substitution transforms the function equation into a simple algebraic equation that we can then manipulate to isolate x. By replacing f(x) with 15, we create a concrete equation that allows us to directly address the unknown variable. This is a fundamental technique in algebra, enabling us to move from a function-based expression to a solvable equation. The act of substitution is not just a mechanical step; it reflects a deep understanding of what a function represents and how it relates to its inputs and outputs. It’s the cornerstone of solving many algebraic problems and provides a clear pathway towards finding the value of x. This initial step is crucial because it sets the stage for all subsequent operations. Without a proper substitution, the equation remains in its functional form, which doesn’t directly reveal the value of x. Therefore, accurately substituting the given value is essential for progressing towards the solution. This step also highlights the importance of reading and understanding the problem statement carefully to ensure that all given information is correctly incorporated into the solution process. By doing so, we establish a solid foundation for the rest of the calculations.

So, we replace f(x) with 15:

15 = -10x + 5

2. Isolate the Term with x

Now that we have the equation 15 = -10x + 5, our next goal is to isolate the term containing x. This means we want to get the term -10x by itself on one side of the equation. To achieve this, we need to eliminate the constant term (+5) from the right side. We can do this by performing the inverse operation, which in this case is subtraction. Subtracting 5 from both sides of the equation maintains the balance and ensures that the equation remains valid. This step is a fundamental application of the properties of equality, which state that performing the same operation on both sides of an equation preserves the equality. Isolating the term with x is a critical step because it simplifies the equation and brings us closer to solving for x. By removing the constant term, we can focus solely on the term that contains the variable we are trying to find. This process is not just about following a mechanical procedure; it’s about understanding the structure of the equation and strategically manipulating it to reveal the solution. Each step is a deliberate move that contributes to the overall goal of isolating x. Moreover, this step demonstrates the importance of performing operations on both sides of the equation to maintain balance and accuracy. It’s a key principle in algebra that underlies many solution strategies. By mastering this technique, we can confidently tackle a wide range of algebraic problems.

To isolate -10x, we subtract 5 from both sides of the equation:

15 - 5 = -10x + 5 - 5

This simplifies to:

10 = -10x

3. Solve for x

After isolating the term -10x, we are now one step closer to solving for x. The equation we have is 10 = -10x. To isolate x completely, we need to eliminate the coefficient -10 from the right side. This can be achieved by performing the inverse operation of multiplication, which is division. Dividing both sides of the equation by -10 will isolate x and reveal its value. This step is a direct application of the properties of equality, ensuring that the solution remains accurate. Solving for x involves not just performing the correct mathematical operation, but also understanding why that operation is necessary. It’s about unraveling the equation and revealing the unknown variable. This final step is often the most satisfying, as it brings the entire solution process to a conclusion. By dividing by the coefficient, we are essentially undoing the multiplication that was initially applied to x. This highlights the inverse relationship between multiplication and division and how they can be used to solve equations. Furthermore, this step underscores the importance of considering the sign of the coefficient when performing division. A negative coefficient requires dividing by a negative number to ensure that the variable is isolated with a positive sign. By mastering this technique, we can confidently solve for variables in various algebraic equations.

To solve for x, we divide both sides of the equation by -10:

10 / -10 = -10x / -10

This simplifies to:

x = -1

Therefore, the solution to the equation f(x) = -10x + 5 given f(x) = 15 is x = -1. This final answer represents the value of x that satisfies both the functional equation and the given condition. To ensure the accuracy of our solution, it's always a good practice to verify the result by plugging it back into the original equation. This verification step helps confirm that our calculations are correct and that the value we found for x indeed makes the equation true. In this case, substituting x = -1 into the equation f(x) = -10x + 5 should yield f(x) = 15. The process of solving for x involves several key steps, including substitution, isolating terms, and performing inverse operations. Each step is crucial in systematically unraveling the equation and arriving at the correct solution. The final answer is not just a number; it’s the culmination of a logical and methodical process. It represents a precise value that fulfills the conditions set forth in the problem. Furthermore, the solution x = -1 can be interpreted graphically as the x-coordinate of the point where the function f(x) = -10x + 5 intersects the horizontal line y = 15. This visual representation can provide a deeper understanding of the solution and its significance. By mastering the techniques used to solve for x, we gain valuable problem-solving skills that can be applied in various mathematical and real-world contexts. This final answer serves as a testament to the power of algebraic manipulation and the importance of a systematic approach to problem-solving.

In summary, we successfully solved for x in the equation f(x) = -10x + 5, given that f(x) = 15. The process involved three main steps: first, substituting the value of f(x); second, isolating the term with x; and third, dividing to solve for x. This systematic approach is fundamental to solving algebraic equations and highlights the importance of understanding inverse operations. By substituting f(x) = 15 into the equation, we transformed the problem into a solvable algebraic form. This substitution is a crucial step, as it bridges the gap between the function notation and a concrete equation that can be manipulated. The subsequent step of isolating the term with x involved subtracting 5 from both sides of the equation, which effectively separated the term containing x from the constant term. This step is a clear demonstration of the properties of equality, where performing the same operation on both sides maintains the balance of the equation. Finally, dividing both sides by -10 allowed us to isolate x completely, revealing the solution x = -1. This final step underscores the inverse relationship between multiplication and division and the importance of considering the sign of the coefficient. Throughout this process, we emphasized the importance of each step and the underlying principles that make it valid. By mastering these techniques, we can confidently approach a wide range of algebraic problems. The solution x = -1 is not just an answer; it’s a result of a logical and methodical process, highlighting the power of algebraic manipulation and the value of a systematic approach to problem-solving.