Solving 5x - 1 = 5(x - 6) A Step-by-Step Guide
Solve the equation 5x-1=5(x-6) for x.
In the realm of mathematics, solving linear equations is a fundamental skill that unlocks the door to a deeper understanding of algebra and its applications. Linear equations, characterized by their straight-line graphs, are ubiquitous in various fields, from physics and engineering to economics and computer science. Mastering the art of solving these equations empowers us to tackle real-world problems with confidence. In this comprehensive guide, we will embark on a journey to solve the equation 5x - 1 = 5(x - 6), meticulously dissecting each step and illuminating the underlying principles. This equation, a classic example of a linear equation with one variable, presents an opportunity to hone our algebraic prowess and develop a systematic approach to problem-solving. By the end of this exploration, you will not only possess the ability to solve this specific equation but also gain a solid foundation for tackling a wide array of linear equations.
Understanding the Equation: Deconstructing 5x - 1 = 5(x - 6)
Before we dive into the solution, let's take a moment to understand the anatomy of the equation 5x - 1 = 5(x - 6). At its core, this equation represents a balance – the left side must equal the right side. The variable 'x' represents an unknown value that we seek to unveil. The equation involves various mathematical operations, including multiplication, subtraction, and the distributive property. To effectively solve for 'x', we must carefully manipulate the equation while preserving its fundamental balance. The left side, 5x - 1, consists of two terms: 5x, which represents 5 times the unknown value 'x', and -1, a constant term. The right side, 5(x - 6), introduces the concept of the distributive property. This property dictates that we must multiply the 5 outside the parentheses by each term inside the parentheses. This initial understanding forms the bedrock upon which we will construct our solution.
The Distributive Property: Expanding the Right Side
The distributive property is a cornerstone of algebraic manipulation, allowing us to simplify expressions involving parentheses. In our equation, 5x - 1 = 5(x - 6), the right side, 5(x - 6), calls for the application of this property. The distributive property states that a(b + c) = ab + ac, where 'a', 'b', and 'c' represent any numbers or variables. Applying this to our equation, we multiply the 5 by both 'x' and -6. This yields 5 * x = 5x and 5 * -6 = -30. Thus, the right side of the equation transforms from 5(x - 6) to 5x - 30. The equation now reads 5x - 1 = 5x - 30. This transformation is a crucial step in isolating 'x', as it eliminates the parentheses and allows us to combine like terms. The distributive property is not just a mathematical trick; it's a fundamental principle that ensures the equation remains balanced as we manipulate it.
Isolating the Variable: Gathering 'x' Terms
Now that we've expanded the right side using the distributive property, our equation stands as 5x - 1 = 5x - 30. The next strategic move is to isolate the variable 'x'. This involves gathering all the terms containing 'x' on one side of the equation. To achieve this, we can subtract 5x from both sides. This operation maintains the equation's balance because we are performing the same action on both sides. Subtracting 5x from the left side, 5x - 1, results in -1. Subtracting 5x from the right side, 5x - 30, results in -30. Our equation now simplifies dramatically to -1 = -30. This seemingly contradictory result is a significant clue that we will unravel in the next section. The process of isolating the variable is a common thread in solving algebraic equations, allowing us to pinpoint the value of the unknown.
Unveiling the Contradiction: Interpreting -1 = -30
After isolating the variable 'x', we arrived at the equation -1 = -30. This statement is clearly a contradiction – negative one does not equal negative thirty. This seemingly nonsensical result is not an error in our calculations; rather, it's a profound revelation about the nature of the original equation. A contradiction indicates that there is no value of 'x' that can satisfy the equation 5x - 1 = 5(x - 6). In other words, the equation has no solution. Graphically, this translates to two parallel lines that never intersect, representing the two sides of the equation. The absence of a solution is a perfectly valid outcome in mathematics, and recognizing such contradictions is a crucial skill. It prevents us from chasing a phantom solution and allows us to correctly interpret the equation's implications.
The Empty Set: Expressing No Solution
In mathematics, the concept of the empty set provides a concise way to express the absence of a solution. The empty set, denoted by the symbol ∅, is a set containing no elements. When an equation has no solution, we can formally state the solution set as the empty set. This is a more rigorous and universally understood way of communicating the absence of a solution compared to simply stating