How Many Solutions Does 4x + 47 = 4(x + 5) Have?

Navigating the realm of algebraic equations often leads us to the crucial question of how many solutions exist. In this article, we embark on a journey to dissect the equation 4x + 47 = 4(x + 5), meticulously exploring its structure and employing algebraic techniques to unveil the number of solutions it possesses. This exploration is not merely an academic exercise; it is a fundamental aspect of mathematical problem-solving, with applications spanning various fields, from engineering and physics to economics and computer science. Understanding the nature of solutions – whether they are singular, nonexistent, or infinite – is paramount in accurately modeling real-world phenomena and making informed decisions.

Dissecting the Equation: A Step-by-Step Analysis

To decipher the solution count, we must first dissect the equation 4x + 47 = 4(x + 5) and meticulously analyze its components. This involves employing the fundamental principles of algebra, such as the distributive property and the manipulation of terms to isolate the variable. Our primary objective is to simplify the equation into a form that reveals the relationship between the variable x and the constants, thereby illuminating the nature of the solutions.

1. Applying the Distributive Property

The initial step in our algebraic expedition involves applying the distributive property to the right side of the equation. The distributive property, a cornerstone of algebraic manipulation, dictates that a(b + c) = ab + ac. In our case, we must distribute the 4 across the terms within the parentheses: 4(x + 5). This yields 4x + 20. Consequently, the equation transforms from its original form to 4x + 47 = 4x + 20. This transformation is a crucial step, as it eliminates the parentheses and brings us closer to isolating the variable x.

2. Isolating the Variable: A Quest for Clarity

With the distributive property applied, we now embark on the quest to isolate the variable x. This involves strategically manipulating the equation to gather all terms containing x on one side and all constant terms on the other. To achieve this, we can subtract 4x from both sides of the equation. This operation maintains the equality of the equation while simultaneously eliminating the x term from the right side. The equation 4x + 47 = 4x + 20 thus metamorphoses into 47 = 20.

3. Unveiling the Truth: A Contradiction Emerges

The equation 47 = 20 presents a stark contradiction. This statement is patently false, irrespective of the value of x. This contradiction is the key to unlocking the solution count. It signifies that there is no value of x that can satisfy the original equation. No matter what number we substitute for x, the equation will never hold true. This realization leads us to a profound conclusion: the equation has no solution.

The Significance of No Solution

The absence of a solution to an equation is not merely a mathematical curiosity; it holds significant implications in various contexts. In real-world modeling, an equation with no solution indicates an inconsistency in the model or a scenario that is impossible. For instance, if we were modeling a physical system and arrived at an equation with no solution, it would suggest that our model is flawed or that the scenario we are trying to represent cannot occur in reality.

Consider a scenario where we are trying to determine the number of hours it would take two people working at different rates to complete a task. If our equation yields no solution, it might indicate that the task cannot be completed within a reasonable timeframe given the rates of the workers. Similarly, in financial modeling, an equation with no solution might suggest that a particular investment strategy is not viable under the given conditions.

Contrasting Scenarios: One Solution and Infinitely Many Solutions

To fully appreciate the significance of an equation having no solution, it is instructive to contrast it with scenarios where an equation has one solution or infinitely many solutions. An equation with one solution, such as 2x + 3 = 7, has a single value of x that satisfies the equation (in this case, x = 2). This represents a specific, unique solution to the problem.

On the other hand, an equation with infinitely many solutions, such as 2x + 4 = 2(x + 2), holds true for all values of x. This occurs when the equation simplifies to an identity, a statement that is always true. In such cases, any value of x will satisfy the equation, indicating a continuous range of solutions.

Equations with One Solution: A Unique Answer

Equations possessing a single, unique solution are ubiquitous in mathematics and its applications. These equations often represent scenarios where a specific value or condition must be met. For example, determining the exact amount of reactants needed for a chemical reaction to proceed completely, calculating the precise trajectory for a spacecraft to reach its destination, or finding the equilibrium price in a market are all instances where a single solution is paramount.

Solving equations with one solution typically involves isolating the variable of interest through algebraic manipulations, such as adding or subtracting terms, multiplying or dividing by constants, and applying inverse operations. The goal is to arrive at an equation of the form x = a, where a is the unique solution.

Equations with Infinitely Many Solutions: A Continuous Spectrum

Equations with infinitely many solutions, while less common than those with a single solution, arise in situations where there is a dependency or redundancy in the variables or conditions. These equations often represent relationships or constraints that hold true across a continuous range of values. For instance, in geometry, the equation of a line represents an infinite set of points that satisfy the linear relationship. In calculus, the indefinite integral of a function represents a family of functions that differ only by a constant, resulting in infinitely many solutions.

Identifying equations with infinitely many solutions involves simplifying the equation to an identity, a statement that is always true regardless of the value of the variable(s). This typically occurs when both sides of the equation are algebraically equivalent.

Back to Our Equation: Solidifying the No Solution Verdict

Returning to our original equation, 4x + 47 = 4(x + 5), our analysis has definitively revealed that it falls into the category of equations with no solution. The contradiction 47 = 20 serves as irrefutable evidence that there is no value of x that can satisfy the equation. This conclusion underscores the importance of rigorous algebraic manipulation and the careful interpretation of results.

Conclusion: The Power of Algebraic Analysis

In conclusion, by meticulously dissecting the equation 4x + 47 = 4(x + 5), we have demonstrated the power of algebraic analysis in determining the number of solutions. Through the application of the distributive property and the strategic isolation of variables, we unveiled a contradiction that definitively proves the absence of a solution. This exploration highlights the fundamental principle that not all equations possess solutions, and understanding the nature of solutions is crucial for accurate mathematical modeling and problem-solving across diverse disciplines. The ability to discern between equations with no solution, one solution, and infinitely many solutions is a cornerstone of mathematical literacy, empowering us to navigate the complexities of the mathematical world with confidence and precision.

Therefore, the equation 4x + 47 = 4(x + 5) has no solution.