Solving 4kx + 10kx = 7 A Step-by-Step Guide
In the realm of algebra, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. This often involves using algebraic manipulations to isolate the variable on one side of the equation. These manipulations include adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity, ensuring the equality remains balanced. Solving equations is not just an abstract exercise; it's a crucial tool in various fields, including physics, engineering, economics, and computer science. Mastering this skill is essential for anyone looking to tackle more advanced mathematical concepts and real-world problems. Different types of equations, such as linear, quadratic, and exponential, require specific methods for solving, adding to the complexity and richness of algebraic problem-solving.
Linear equations, the simplest form, involve variables raised to the first power and can typically be solved using basic algebraic operations. Quadratic equations, which include variables raised to the second power, often require factoring, completing the square, or using the quadratic formula. More complex equations may involve radicals, rational expressions, or trigonometric functions, each necessitating unique strategies and techniques. Understanding the underlying principles of equation solving provides a strong foundation for advanced mathematical studies and practical applications. Moreover, the ability to solve equations efficiently and accurately is highly valued in various professional domains, making it a crucial skill for students and professionals alike. In the context of problem-solving, equations serve as models that represent real-world situations, allowing us to find solutions to practical problems by manipulating and solving these equations.
Let's address the problem at hand: solving the equation 4kx + 10kx = 7 for x. This equation is a linear equation in terms of x, where k is a constant. Our primary objective is to isolate x on one side of the equation to determine its value. This involves applying algebraic principles to simplify the equation and perform the necessary operations. The presence of the constant k adds a layer of abstraction, but the fundamental approach remains the same: we aim to manipulate the equation while maintaining its balance, ultimately solving for x in terms of k. Before diving into the step-by-step solution, it's important to recognize the structure of the equation and identify the terms involving x. By combining like terms and applying the properties of equality, we can systematically work towards isolating x and finding its value. This problem serves as a good illustration of how algebraic techniques can be used to solve equations involving variables and constants, a common scenario in mathematical and scientific problem-solving.
This equation is a straightforward example of a linear equation, but it's important to approach it systematically to avoid common pitfalls. A clear and methodical approach is essential to accurately solve this equation and express the solution in terms of k. Recognizing the linearity of the equation allows us to apply standard algebraic techniques, ensuring a clear and logical progression towards the solution. Furthermore, this type of problem reinforces the importance of careful manipulation and attention to detail when working with algebraic expressions. The ability to solve such equations is a fundamental skill in algebra and is crucial for tackling more complex problems in mathematics and related fields. In the following sections, we'll break down the solution step-by-step, highlighting the key principles and techniques involved in isolating x and finding its value.
Step-by-Step Solution
1. Combine Like Terms
The first step in solving the equation 4kx + 10kx = 7 is to combine the like terms on the left-hand side. Both terms, 4kx and 10kx, contain the variable x and the constant k, making them like terms. We can combine them by adding their coefficients: 4k + 10k = 14k. Therefore, the equation simplifies to 14kx = 7. This step is crucial as it reduces the number of terms and makes the equation easier to manipulate. Combining like terms is a fundamental technique in algebra, allowing us to simplify expressions and equations by grouping together terms with the same variables and powers. This process not only makes the equation more manageable but also helps in identifying the structure and properties of the equation. By simplifying the equation in this manner, we set the stage for the next steps in isolating the variable x and finding its value. This initial simplification is a key component of effective algebraic problem-solving and is often the first step in tackling more complex equations.
Combining like terms is a basic yet essential algebraic operation that streamlines the process of solving equations. It involves adding or subtracting terms that have the same variable raised to the same power. In this specific case, both 4kx and 10kx have the variable x raised to the power of 1, making them like terms. The coefficients 4k and 10k can be added together, resulting in 14k. This simplification is crucial for isolating the variable and solving the equation. Recognizing and combining like terms is a fundamental skill in algebra and is a prerequisite for solving more complex equations and systems of equations. The ability to efficiently combine like terms reduces the complexity of an equation and allows for a clearer understanding of the relationships between the variables and constants involved. This step is often the first in a series of algebraic manipulations aimed at finding the solution to an equation.
2. Isolate x
Now that we have the simplified equation 14kx = 7, the next step is to isolate x. To do this, we need to get x by itself on one side of the equation. Since x is being multiplied by 14k, we can isolate x by dividing both sides of the equation by 14k. This gives us x = 7 / (14k). Dividing both sides of the equation by the same non-zero quantity maintains the equality, which is a fundamental principle in solving equations. This step effectively separates x from the constant term, bringing us closer to finding the solution. Isolating the variable is a critical technique in algebra, and it is often the key step in solving equations of various types. By performing this division, we have successfully isolated x, expressing it in terms of the constant k. This process highlights the importance of inverse operations in algebra, where multiplication is undone by division, and vice versa.
Isolating the variable is the core concept in solving equations. This involves performing algebraic operations to get the variable alone on one side of the equation. In the equation 14kx = 7, x is multiplied by 14k. To isolate x, we perform the inverse operation, which is division. Dividing both sides of the equation by 14k maintains the equality while effectively separating x from the constant term. This step is crucial as it expresses x in terms of other constants and variables, ultimately leading to the solution. The process of isolating a variable is a cornerstone of algebraic problem-solving and is applicable to a wide range of equations, from simple linear equations to more complex equations involving multiple variables and operations. By mastering this technique, one can solve for any variable in an equation, provided that enough information is given.
3. Simplify the Fraction
We now have x = 7 / (14k). To simplify this fraction, we can observe that both the numerator (7) and the denominator (14k) have a common factor of 7. Dividing both the numerator and the denominator by 7, we get x = 1 / (2k). Simplifying fractions is an essential step in presenting the solution in its most concise and understandable form. By reducing the fraction to its lowest terms, we eliminate common factors and make the expression simpler to work with. This step not only makes the solution more elegant but also can reveal further insights or patterns. In this case, simplifying the fraction makes the relationship between x and k more apparent. Fraction simplification is a fundamental skill in arithmetic and algebra, and it is often necessary to perform this step to arrive at the final, most simplified solution.
Simplifying fractions is a crucial skill in mathematics, ensuring that the answer is presented in its most reduced form. In the equation x = 7 / (14k), both the numerator and denominator share a common factor of 7. Dividing both by 7 results in the simplified fraction x = 1 / (2k). This simplification makes the solution cleaner and easier to interpret. Simplifying fractions not only reduces the numbers involved but also can reveal underlying patterns and relationships within the equation. This skill is essential for problem-solving in various mathematical contexts and is a fundamental aspect of mathematical proficiency. A simplified fraction is easier to work with in subsequent calculations and provides a clearer representation of the relationship between the variables and constants involved.
Answer
The solution to the equation 4kx + 10kx = 7 for x is x = 1 / (2k).
Selecting the Correct Option
Comparing our solution x = 1 / (2k) with the given options:
- A. x = 1 / (2k)
- B. x = 2k
- C. x = k / 2
- D. x = 2 / k
We can see that option A, x = 1 / (2k), matches our solution. Therefore, the correct answer is option A.
In conclusion, solving the equation 4kx + 10kx = 7 involves combining like terms, isolating the variable x, and simplifying the resulting expression. The step-by-step approach we followed demonstrates how basic algebraic principles can be applied to solve linear equations effectively. The solution, x = 1 / (2k), is obtained by first combining the terms 4kx and 10kx to get 14kx = 7. Then, dividing both sides by 14k isolates x, resulting in x = 7 / (14k). Finally, simplifying the fraction by dividing both numerator and denominator by 7 gives the final answer, x = 1 / (2k). This exercise highlights the importance of methodical problem-solving in algebra and reinforces the fundamental techniques of equation manipulation. Understanding these techniques is crucial for solving more complex mathematical problems and for applications in various scientific and engineering fields. Moreover, this problem illustrates how a seemingly complex equation can be solved through a series of logical and straightforward steps.
Solving algebraic equations is a fundamental skill in mathematics, and the process demonstrated here provides a clear pathway for approaching such problems. By breaking down the equation into manageable steps, we can systematically isolate the variable and find its value. This approach not only leads to the correct answer but also enhances understanding of the underlying principles of algebra. The ability to solve equations is essential for various applications in science, engineering, economics, and other fields. Therefore, mastering these basic techniques is crucial for anyone pursuing studies or careers in these areas. The problem-solving strategy outlined here can be applied to a wide range of equations, making it a valuable tool for mathematical problem-solving. The final solution, x = 1 / (2k), demonstrates the relationship between x and the constant k, providing insights into the behavior of the equation.
