Solving Equations A Step-by-Step Guide To X + 4(x + 5) = 40

In the realm of mathematics, equations serve as puzzles, challenging us to decipher the hidden value of an unknown variable. Today, we embark on a journey to solve the equation x + 4(x + 5) = 40, a seemingly simple yet intriguing expression that holds the key to a specific numerical solution. Our mission is to meticulously dissect this equation, employing the fundamental principles of algebra to isolate the variable x and ultimately unveil its true value. We will navigate through the order of operations, the distributive property, and the art of combining like terms, all in pursuit of the correct answer from the given options: A. x = 4, B. x = 9, C. x = 12, and D. x = 7. So, let's delve into the heart of this equation and embark on a step-by-step exploration to uncover its hidden solution.

Dissecting the Equation: A Step-by-Step Approach

To begin our quest for the solution, we must first understand the anatomy of the equation. The equation x + 4(x + 5) = 40 presents us with a variable x, several constants, and mathematical operations that bind them together. The left-hand side of the equation, x + 4(x + 5), is an expression that needs simplification, while the right-hand side, 40, is a constant value. Our goal is to manipulate the left-hand side in such a way that we isolate x and determine its numerical value. To achieve this, we will employ a series of algebraic techniques, carefully following the order of operations to maintain the equation's balance.

The first step in simplifying the equation is to address the parentheses. The term 4(x + 5) signifies that the entire expression within the parentheses, (x + 5), is multiplied by 4. This is where the distributive property comes into play. The distributive property states that a(b + c) = ab + ac. Applying this property to our equation, we multiply 4 by both x and 5, resulting in 4x + 20. Now, our equation transforms into x + 4x + 20 = 40. This seemingly small step has expanded the equation, revealing more terms that we can work with.

With the parentheses resolved, our next task is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we have two terms containing the variable x: x and 4x. These terms can be combined by adding their coefficients. The coefficient of x is implicitly 1, so x + 4x becomes (1 + 4)x, which simplifies to 5x. Our equation now reads 5x + 20 = 40. We have successfully reduced the number of terms on the left-hand side, bringing us closer to isolating x.

Now, our focus shifts to isolating the term containing x. To do this, we need to eliminate the constant term 20 from the left-hand side. The principle of equality dictates that we can perform the same operation on both sides of the equation without altering its balance. To eliminate 20, we subtract 20 from both sides of the equation. This gives us 5x + 20 - 20 = 40 - 20, which simplifies to 5x = 20. We have successfully isolated the term 5x on the left-hand side.

Finally, to isolate x completely, we need to get rid of the coefficient 5. Since 5 is multiplying x, we perform the inverse operation, which is division. We divide both sides of the equation by 5, resulting in (5x) / 5 = 20 / 5. This simplifies to x = 4. We have successfully solved for x, and the solution is 4.

Verifying the Solution: Ensuring Accuracy

In the world of mathematics, accuracy is paramount. Once we arrive at a solution, it's crucial to verify its correctness. To verify our solution x = 4, we substitute this value back into the original equation and check if the equation holds true. Substituting x = 4 into x + 4(x + 5) = 40, we get 4 + 4(4 + 5) = 40. Now, we simplify the left-hand side following the order of operations.

First, we address the parentheses: 4 + 5 = 9. Our equation becomes 4 + 4(9) = 40. Next, we perform the multiplication: 4 * 9 = 36. The equation now reads 4 + 36 = 40. Finally, we perform the addition: 4 + 36 = 40. The left-hand side simplifies to 40, which is equal to the right-hand side. This confirms that our solution x = 4 is indeed correct. The equation holds true when x = 4, providing us with the assurance that we have successfully unraveled the puzzle.

The Answer Revealed: x = 4

After our meticulous step-by-step exploration and verification, we have arrived at the solution to the equation x + 4(x + 5) = 40. The value of x that satisfies this equation is 4. Therefore, the correct answer from the given options is A. x = 4. Our journey through the algebraic landscape has led us to a definitive conclusion, showcasing the power of mathematical principles in solving equations.

Key Concepts Revisited:

• Order of Operations: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to ensure calculations are performed in the correct sequence.
• Distributive Property: a(b + c) = ab + ac is crucial for expanding expressions involving parentheses.
• Combining Like Terms: Simplifying equations by grouping terms with the same variable and exponent.
• Principle of Equality: Performing the same operation on both sides of an equation maintains balance and allows for manipulation towards isolating the variable.
• Verification: Always substitute the solution back into the original equation to confirm its validity.

Understanding and applying these concepts are fundamental to solving algebraic equations effectively and confidently. The ability to dissect an equation, apply appropriate techniques, and verify the solution is a valuable skill in mathematics and beyond.

Exploring the Significance of Equations in Mathematics

Equations, like the one we've just solved, are the very backbone of mathematics. They are the language through which we describe relationships between quantities, model real-world phenomena, and solve problems across diverse fields. From the simplest arithmetic expressions to the most complex scientific formulas, equations provide a framework for understanding and manipulating the world around us. Understanding what is the solution to this equation is a core skill in mathematics.

The power of an equation lies in its ability to represent a balance, a state of equilibrium between two expressions. The equals sign (=) acts as a fulcrum, indicating that the values on either side are equivalent. This concept of balance is fundamental to mathematical reasoning. When we manipulate an equation, we are essentially performing operations that maintain this balance, ensuring that the relationship between the quantities remains consistent. For example, when we solve for x, x + 4(x + 5) = 40, we use this concept.

Equations serve as a bridge between the abstract world of mathematical symbols and the concrete world of real-life applications. They allow us to translate problems into a symbolic form, apply mathematical techniques to find solutions, and then interpret those solutions back in the context of the original problem. This process of mathematical modeling is crucial in fields such as physics, engineering, economics, and computer science.

Consider, for instance, the equation F = ma, a cornerstone of Newtonian physics. This simple equation encapsulates the relationship between force (F), mass (m), and acceleration (a). It allows us to predict the motion of objects under the influence of forces, from the trajectory of a baseball to the orbit of a planet. The ability to express such fundamental relationships in the form of equations is what makes mathematics such a powerful tool for scientific inquiry.

Equations also play a vital role in solving practical problems in our daily lives. From calculating the cost of groceries to determining the optimal route for a journey, we often rely on equations, either explicitly or implicitly, to make informed decisions. The ability to set up and solve equations is a fundamental skill for navigating the complexities of modern life. Therefore, finding the solution to this equation is something that would be used in real-life situations.

The equation x + 4(x + 5) = 40 that we tackled in this article is a linear equation, a type of equation that forms a straight line when graphed. Linear equations are among the simplest and most widely used types of equations in mathematics. They arise in numerous contexts, from modeling linear relationships between variables to solving systems of equations. Mastering the techniques for solving linear equations is a crucial stepping stone to understanding more advanced mathematical concepts.

The process of solving an equation is not merely a mechanical exercise; it is a journey of discovery. It requires us to think critically, to analyze the structure of the equation, and to strategically apply algebraic techniques. Each step we take in solving an equation brings us closer to the solution, unveiling the hidden value of the unknown variable. The sense of accomplishment that comes from solving a challenging equation is a testament to the power of mathematical thinking.

In conclusion, equations are the lifeblood of mathematics, providing a language for expressing relationships, modeling phenomena, and solving problems. The equation x + 4(x + 5) = 40 serves as a microcosm of this broader significance, demonstrating the power of algebraic techniques in unraveling mathematical puzzles. As we continue our mathematical journey, we will encounter equations of ever-increasing complexity, but the fundamental principles we have learned here will serve as a solid foundation for our explorations.

Expanding Our Mathematical Toolkit: Beyond Linear Equations

While linear equations form a cornerstone of algebra, the mathematical landscape extends far beyond them. As we advance in our mathematical journey, we encounter a diverse array of equation types, each with its own unique characteristics and solution techniques. Exploring these different types of equations expands our mathematical toolkit, empowering us to tackle a wider range of problems.

Quadratic equations, for example, introduce a new level of complexity. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero. The presence of the x² term introduces a non-linear element, leading to solutions that may be real or complex numbers. Solving quadratic equations often involves techniques such as factoring, completing the square, or using the quadratic formula.

Beyond quadratic equations lie polynomial equations of higher degrees. These equations, involving terms with x raised to powers greater than 2, can be significantly more challenging to solve. Techniques such as the rational root theorem and synthetic division can be employed to find rational solutions, but in many cases, numerical methods are required to approximate the solutions.

Equations also appear in the realm of trigonometry, where we encounter trigonometric functions such as sine, cosine, and tangent. Trigonometric equations involve finding angles that satisfy specific relationships involving these functions. Solving trigonometric equations often requires knowledge of trigonometric identities and the periodic nature of trigonometric functions.

Exponential and logarithmic equations introduce another layer of complexity. Exponential equations involve variables in the exponent, while logarithmic equations involve logarithms of variables. Solving these equations often requires using the properties of exponents and logarithms to isolate the variable. Therefore, when looking at the solution to this equation, remember there are many other types of equations to consider.

Systems of equations involve multiple equations with multiple variables. Solving a system of equations requires finding values for the variables that satisfy all the equations simultaneously. Techniques such as substitution, elimination, and matrix methods can be used to solve systems of equations.

The diversity of equation types reflects the richness and breadth of mathematics. Each type of equation presents its own unique challenges and opportunities for exploration. As we learn to navigate this landscape, we develop a deeper appreciation for the power and versatility of mathematics.

The equation x + 4(x + 5) = 40 that we solved in this article serves as a starting point for this broader exploration. The skills and concepts we have learned in solving this equation provide a foundation for tackling more complex mathematical challenges. By expanding our mathematical toolkit and embracing the diversity of equation types, we can unlock new insights and solve problems in a wide range of fields. Remember, understanding the solution to this equation is the starting point of a much larger mathematical world.

The Beauty and Elegance of Mathematical Problem-Solving

In the realm of mathematics, problem-solving is not merely a technical exercise; it is an art form. The process of dissecting a problem, identifying the key concepts, and applying appropriate techniques to arrive at a solution can be a deeply satisfying experience. The beauty of mathematical problem-solving lies in its elegance, its precision, and its ability to reveal hidden truths about the world around us.

The equation x + 4(x + 5) = 40 that we solved in this article exemplifies this elegance. The equation itself is a concise statement of a relationship between quantities. The solution process, involving a series of logical steps, unfolds like a carefully choreographed dance. Each step builds upon the previous one, leading us inexorably towards the answer. The final solution, x = 4, is a simple yet profound statement, encapsulating the essence of the problem.

The elegance of mathematical problem-solving is not limited to simple equations. It extends to the most complex mathematical problems, from Fermat's Last Theorem to the Riemann Hypothesis. These problems, which have baffled mathematicians for centuries, are testaments to the enduring power of mathematical inquiry.

The beauty of mathematical problem-solving also lies in its precision. Mathematics is a language of certainty, where every statement must be rigorously justified. There is no room for ambiguity or approximation. The solution to a mathematical problem is either right or wrong, and the process of arriving at the solution must be logically sound.

This emphasis on precision fosters a sense of intellectual honesty and rigor. It teaches us to think clearly, to express ourselves precisely, and to defend our conclusions with evidence. These skills are valuable not only in mathematics but also in all aspects of life.

The satisfaction that comes from solving a mathematical problem is often a result of the intellectual challenge it presents. A challenging problem forces us to think creatively, to explore different approaches, and to persevere in the face of difficulty. The moment of insight, when the solution suddenly becomes clear, is a moment of pure intellectual joy. Unlocking the solution to this equation is a testament to mathematical understanding.

Mathematical problem-solving is not just about finding answers; it is about developing a way of thinking. It teaches us to approach problems systematically, to break them down into smaller parts, and to identify patterns and relationships. These problem-solving skills are transferable to a wide range of situations, from designing a building to managing a business.

In conclusion, mathematical problem-solving is a beautiful and elegant art form. It combines logic, creativity, and precision to reveal hidden truths and solve complex problems. The equation x + 4(x + 5) = 40 that we solved in this article serves as a small example of this larger world of mathematical beauty and elegance. As we continue our mathematical journey, let us embrace the challenges, celebrate the insights, and appreciate the beauty of mathematical problem-solving. Therefore, understanding the solution to this equation can help in the further understanding of mathematics as a whole.