In this article, we will delve into the process of solving a fundamental algebraic equation: 1/4x + 12 = -1/4x. This equation falls under the category of linear equations, which are a cornerstone of algebra and have widespread applications in various fields, including mathematics, physics, engineering, and economics. Linear equations involve variables raised to the power of one, and solving them entails isolating the variable to determine its value. Understanding how to solve linear equations is crucial for building a solid foundation in mathematics and for tackling more complex problems in the future. We will explore the step-by-step method to solve this particular equation, providing a clear and comprehensive explanation of each operation involved. This will not only help in understanding the solution to this specific equation but also provide a framework for solving similar linear equations.
Before we dive into solving the equation, let's briefly discuss what linear equations are. 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 raised to the power of one, and there are no exponents or other complex functions involved. Linear equations can be represented graphically as a straight line on a coordinate plane. The general form of a linear equation is ax + b = c, where x is the variable, and a, b, and c are constants. Solving a linear equation means finding the value of the variable x that makes the equation true. This is achieved by performing operations on both sides of the equation to isolate the variable on one side. The operations must maintain the equality of the equation, meaning whatever is done to one side must also be done to the other side. The principles of addition, subtraction, multiplication, and division are fundamental in solving linear equations. By understanding these principles, we can systematically manipulate the equation to arrive at the solution. Linear equations are not just theoretical concepts; they are used extensively in real-world applications to model relationships between quantities, such as distance and time, cost and quantity, or supply and demand. Therefore, mastering the techniques for solving linear equations is an essential skill for anyone pursuing studies or careers in STEM fields or economics.
To solve the equation 1/4x + 12 = -1/4x, we need to isolate the variable x on one side of the equation. Here’s a step-by-step breakdown of the process:
Step 1: Combine the x terms
The first step is to gather all the terms containing the variable x on one side of the equation. In this case, we have 1/4x on the left side and -1/4x on the right side. To combine these terms, we can add 1/4x to both sides of the equation. This will eliminate the x term from the right side:
1/4x + 12 + 1/4x = -1/4x + 1/4x
Simplifying this, we get:
(1/4x + 1/4x) + 12 = 0
Combining the fractions, 1/4x + 1/4x equals 2/4x, which simplifies to 1/2x. So the equation becomes:
1/2x + 12 = 0
This step is crucial as it brings all the terms involving the variable to one side, making it easier to isolate the variable in the subsequent steps. The key principle here is the addition property of equality, which states that adding the same quantity to both sides of an equation preserves the equality. By adding 1/4x to both sides, we maintain the balance of the equation while progressing towards isolating x.
Step 2: Isolate the x term
Now that we have 1/2x + 12 = 0, our next goal is to isolate the term containing x. To do this, we need to eliminate the constant term, which is 12, from the left side of the equation. We can achieve this by subtracting 12 from both sides of the equation:
1/2x + 12 - 12 = 0 - 12
Simplifying, we get:
1/2x = -12
This step is another application of the properties of equality. Specifically, we use the subtraction property of equality, which states that subtracting the same quantity from both sides of an equation preserves the equality. By subtracting 12 from both sides, we isolate the term with x on the left side, bringing us closer to solving for x. The equation 1/2x = -12 now shows a direct relationship between x and a constant, making the final step of solving for x more straightforward.
Step 3: Solve for x
We now have the equation 1/2x = -12. To solve for x, we need to get x by itself. Since x is being multiplied by 1/2, we can undo this multiplication by multiplying both sides of the equation by the reciprocal of 1/2, which is 2:
2 * (1/2x) = 2 * (-12)
On the left side, 2 multiplied by 1/2x simplifies to x:
x = 2 * (-12)
On the right side, 2 multiplied by -12 equals -24:
x = -24
Therefore, the solution to the equation 1/4x + 12 = -1/4x is x = -24. This final step utilizes the multiplication property of equality, which states that multiplying both sides of an equation by the same non-zero quantity preserves the equality. By multiplying both sides by 2, we effectively isolated x and found its value. The solution x = -24 is the value that, when substituted back into the original equation, will make the equation true. This can be verified by substituting -24 for x in the original equation and checking if both sides are equal.
To ensure our solution is correct, we can substitute x = -24 back into the original equation: 1/4x + 12 = -1/4x. Substituting -24 for x, we get:
1/4 * (-24) + 12 = -1/4 * (-24)
Now, we simplify each side of the equation:
On the left side:
1/4 * (-24) = -6
So, the left side becomes:
-6 + 12 = 6
On the right side:
-1/4 * (-24) = 6
Thus, the equation becomes:
6 = 6
Since both sides of the equation are equal, our solution x = -24 is correct. This verification step is crucial in problem-solving as it confirms the accuracy of the solution and helps identify any potential errors made during the solving process. By substituting the solution back into the original equation, we ensure that the value we found for x satisfies the equation's conditions. This practice reinforces the understanding of equation solving and enhances confidence in the solution.
In this article, we successfully solved the linear equation 1/4x + 12 = -1/4x and found the solution to be x = -24. We accomplished this by systematically applying the properties of equality to isolate the variable x. The steps involved combining like terms, isolating the x term, and then solving for x. We also verified our solution by substituting it back into the original equation, confirming its accuracy. Understanding how to solve linear equations is a fundamental skill in algebra, and this example demonstrates the process clearly. The principles and techniques discussed here can be applied to solve a wide range of linear equations. Linear equations are a basic yet essential tool in mathematics and its applications. Mastering the process of solving them lays the groundwork for understanding more advanced mathematical concepts and solving real-world problems. The ability to manipulate equations and isolate variables is a valuable skill that extends beyond mathematics and into various fields of science, engineering, and economics. By practicing and understanding these fundamental principles, individuals can build confidence in their mathematical abilities and approach problem-solving with a systematic and logical approach.
