Finding The Value Of A In Function F(x) = (x+15)/5

In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between variables. The given problem presents us with a linear function, f(x) = (x+15)/5, and a specific condition: f(a) = 10. Our mission is to unravel the value of the constant a that satisfies this condition. This exploration will not only demonstrate the application of algebraic principles but also highlight the elegance and power of functions in solving mathematical puzzles. We will delve into the mechanics of function evaluation, equation solving, and the significance of constants within mathematical expressions. This journey will strengthen your understanding of functions and their role in problem-solving scenarios.

Understanding the Function

The function f(x) = (x+15)/5 is a linear function. This means that if we were to plot it on a graph, it would form a straight line. The function takes an input x, adds 15 to it, and then divides the result by 5. This process defines the output of the function for any given input. Understanding the structure of the function is crucial because it lays the foundation for solving problems related to it. For instance, if we want to find the output for a specific input, we simply substitute the input value for x in the function's equation. Conversely, if we know the output and want to find the corresponding input, we need to solve the equation for x. In our case, we know that f(a) = 10, which means when we input a into the function, the output is 10. This provides us with the necessary information to determine the value of a.

Setting up the Equation

Our key to unlocking the value of a lies in the given condition: f(a) = 10. This tells us that when we substitute a for x in the function f(x), the result is 10. Therefore, we can set up the equation:

(a + 15) / 5 = 10

This equation represents the heart of the problem. It directly links the unknown value a to the known output of the function, 10. Solving this equation will reveal the value of a that satisfies the given condition. The equation embodies the relationship defined by the function and the specific scenario presented in the problem. By manipulating this equation using algebraic principles, we can isolate a and determine its value. The process of setting up the equation is a crucial step in solving mathematical problems, as it translates the problem's conditions into a form that can be solved using mathematical tools.

Solving for a

Now, let's embark on the journey of solving the equation (a + 15) / 5 = 10. Our goal is to isolate a on one side of the equation. To achieve this, we'll employ a series of algebraic manipulations. The first step involves undoing the division by 5. We can accomplish this by multiplying both sides of the equation by 5:

5 * ((a + 15) / 5) = 10 * 5

This simplifies to:

a + 15 = 50

Next, we need to isolate a further by removing the addition of 15. We can do this by subtracting 15 from both sides of the equation:

a + 15 - 15 = 50 - 15

This leads us to the solution:

a = 35

Therefore, the value of a that satisfies the condition f(a) = 10 is 35. This meticulous step-by-step process demonstrates the power of algebraic manipulation in solving equations and extracting unknown values. Each step is a logical progression, building upon the previous one, to ultimately reveal the solution.

Verifying the Solution

In mathematics, it's always a good practice to verify our solutions. This ensures that we haven't made any errors in our calculations and that our answer indeed satisfies the original conditions of the problem. To verify our solution, a = 35, we'll substitute this value back into the original function and see if we get the expected output, 10:

f(35) = (35 + 15) / 5

f(35) = 50 / 5

f(35) = 10

As we can see, when we substitute a = 35 into the function, the output is indeed 10. This confirms that our solution is correct. Verification is a crucial step in the problem-solving process, as it provides confidence in the accuracy of our answer and helps us identify any potential errors.

The Answer

Having successfully solved the equation and verified our solution, we can confidently state that the value of a is 35. This corresponds to option C in the given choices. This final answer represents the culmination of our problem-solving journey, where we started with a function and a condition and ended with the specific value that satisfies that condition.

In this exploration, we've successfully determined the value of a in the function f(x) = (x+15)/5 given the condition f(a) = 10. We navigated through the process of understanding the function, setting up the equation, solving for a, and verifying our solution. This exercise not only demonstrates the practical application of functions and algebraic principles but also underscores the importance of meticulousness and verification in mathematical problem-solving. The problem highlights the elegance and power of mathematical tools in unraveling seemingly complex scenarios. This journey has reinforced our understanding of functions and their role in representing relationships between variables and solving real-world problems. By mastering these fundamental concepts, we can confidently tackle a wide range of mathematical challenges.

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This problem serves as a great example of how functions can be used to model relationships and solve for unknowns. It also highlights the importance of understanding the basic operations of algebra, such as substitution, simplification, and solving equations. The process of verifying the solution is a valuable habit to develop, as it helps to ensure accuracy and build confidence. Similar problems can be encountered in various fields, including physics, engineering, and economics, where functions are used to model real-world phenomena. This problem serves as a foundational stepping stone for more advanced mathematical concepts and applications.

To further solidify your understanding, try solving similar problems with different functions and conditions. For instance:

1. If g(x) = (2x - 7) / 3 and g(b) = 5, find the value of b.
2. If h(x) = 4x + 9 and h(c) = 21, find the value of c.
3. If k(x) = (x / 2) - 3 and k(d) = -1, find the value of d.

These practice problems will allow you to apply the same problem-solving strategies we used in this example and further enhance your understanding of functions and equation solving. Remember to always verify your solutions to ensure accuracy.