Solving The Equation 5(x-6)+3=-7 A Step-by-Step Guide

In the intricate world of mathematics, equations serve as fundamental building blocks, representing relationships between variables and constants. Solving equations is akin to deciphering a code, where the goal is to unveil the value of the unknown variable that satisfies the given equation. The equation presented, 5(x-6)+3=-7, is a linear equation, a type of equation that involves a single variable raised to the power of one. Linear equations are ubiquitous in mathematics and its applications, making their solution a crucial skill to master.

Solving linear equations involves a systematic approach, guided by the principles of algebraic manipulation. The core strategy is to isolate the variable on one side of the equation, effectively separating it from the constants and other terms. This isolation is achieved by performing a series of operations on both sides of the equation, ensuring that the equality remains balanced. The operations employed are dictated by the structure of the equation, aiming to undo the operations that bind the variable.

Let's embark on the journey of solving the equation 5(x-6)+3=-7, meticulously charting each step along the way. Our initial task is to simplify the equation, paving the path for isolating the variable x. The first obstacle is the parentheses, which we conquer by applying the distributive property. This property dictates that multiplying a sum or difference by a number is equivalent to multiplying each term inside the parentheses by that number. In our case, we distribute the 5 across the terms within the parentheses, yielding 5 * x - 5 * 6, which simplifies to 5x - 30. Our equation now transforms into 5x - 30 + 3 = -7.

The quest for simplification continues as we combine the constant terms on the left side of the equation. We have -30 and +3, which, when combined, result in -27. Our equation now takes the form 5x - 27 = -7. We are steadily progressing towards isolating x, the elusive variable we seek to unveil.

To isolate x, we must first eliminate the constant term, -27, from the left side of the equation. We achieve this by performing the inverse operation, which is adding 27 to both sides of the equation. This maneuver ensures that the equality remains intact, while strategically shifting the constant term to the right side. Adding 27 to both sides of 5x - 27 = -7 yields 5x = 20.

The final step in our quest is to isolate x completely. It is currently bound to the coefficient 5, which multiplies it. To undo this multiplication, we perform the inverse operation, which is dividing both sides of the equation by 5. This action effectively separates x from its coefficient, revealing its true value. Dividing both sides of 5x = 20 by 5 gives us x = 4. We have successfully deciphered the equation, unearthing the solution x = 4.

To ensure the accuracy of our solution, we perform a crucial step: verification. We substitute the value we found, x = 4, back into the original equation, 5(x-6)+3=-7, and check if the equality holds true. Replacing x with 4, we get 5(4-6)+3=-7. Simplifying the expression inside the parentheses, we have 5(-2)+3=-7. Multiplying 5 by -2, we obtain -10+3=-7. Finally, adding -10 and 3, we arrive at -7=-7. The equality holds true, confirming that our solution, x = 4, is indeed correct.

Therefore, the solution to the equation 5(x-6)+3=-7 is A. x=4. We have successfully navigated the realm of algebraic equations, employing the principles of simplification and isolation to unveil the value of the unknown variable.

Mastering the Art of Equation Solving

Equation solving is a fundamental skill in mathematics, essential for tackling a wide array of problems across various disciplines. Mastering equation solving requires a deep understanding of algebraic principles and a systematic approach to problem-solving. While the process may seem daunting at first, with practice and guidance, it becomes an intuitive and rewarding endeavor.

Beyond Linear Equations: Exploring the Landscape

While linear equations form the bedrock of equation solving, the mathematical landscape extends far beyond. Quadratic equations, with their variable raised to the power of two, introduce a new level of complexity. Cubic equations, with the variable raised to the power of three, further expand the possibilities. Each type of equation demands its own set of techniques and strategies for solution.

The Power of Equations: Real-World Applications

Equations are not merely abstract mathematical constructs; they are powerful tools that find applications in countless real-world scenarios. From physics and engineering to economics and finance, equations serve as the language for describing and modeling phenomena. They enable us to predict outcomes, optimize processes, and make informed decisions.

Equations in Physics: Unraveling the Laws of Nature

In the realm of physics, equations are the cornerstone of understanding the laws of nature. Newton's laws of motion, Maxwell's equations of electromagnetism, and Einstein's theory of relativity are all expressed in the form of equations. These equations allow physicists to describe the motion of objects, the behavior of electromagnetic fields, and the nature of gravity.

Equations in Engineering: Designing and Building the World

Engineers rely heavily on equations to design and build the world around us. Structural engineers use equations to calculate the forces acting on bridges and buildings, ensuring their stability. Electrical engineers use equations to design circuits and electronic devices. Chemical engineers use equations to model chemical reactions and design industrial processes.

Equations in Economics and Finance: Navigating the Markets

Economists and financial analysts use equations to model economic systems and financial markets. Supply and demand curves, interest rate models, and stock pricing formulas are all expressed as equations. These equations help economists and analysts understand economic trends, predict market movements, and make investment decisions.

In conclusion, equations are an integral part of mathematics and its applications. From simple linear equations to complex systems of equations, they provide a framework for understanding and solving problems across a wide range of disciplines. Mastering equation solving is a valuable skill that empowers individuals to navigate the complexities of the world and make informed decisions.