Solving 78x - 2x + 15y - X + 3x = 360 + 29 - 5 + 6 A Step-by-Step Guide
Hey guys! Let's dive into the world of algebra and tackle a common type of problem: solving equations. Today, we're going to break down the equation 78x - 2x + 15y - x + 3x = 360 + 29 - 5 + 6 step by step. Don't worry if it looks intimidating at first; we'll make it super clear and easy to understand. We will focus on simplifying expressions, combining like terms, and isolating variables. These concepts form the bedrock of algebra and are vital for tackling more complex mathematical problems. So, grab your calculators and let's get started! We'll transform this seemingly complex equation into a manageable form, demonstrating the power and elegance of algebraic manipulation. By the end of this guide, you'll not only understand how to solve this specific equation but also gain a deeper appreciation for the fundamental principles that govern algebraic problem-solving. This foundational knowledge will empower you to confidently tackle a wide range of mathematical challenges, both in academic settings and real-world applications. Let's embark on this algebraic journey together and unlock the secrets hidden within this equation!
1. Understanding the Basics of Algebraic Equations
Before we jump into solving the equation, let's quickly recap some fundamental concepts. In algebra, an equation is a statement that two expressions are equal. Our goal is usually to find the value(s) of the variable(s) that make the equation true. In our case, we have two variables, x and y. Solving equations often involves a series of manipulations to isolate the variable we're interested in. This might involve combining like terms, which means adding or subtracting terms with the same variable and exponent. For example, 2x and 3x are like terms, but 2x and 2x² are not. We also use the properties of equality, which state that we can perform the same operation on both sides of the equation without changing the solution. This includes adding, subtracting, multiplying, or dividing both sides by the same non-zero number. Understanding these basics is crucial because it forms the foundation upon which we build our problem-solving strategy. We'll apply these principles throughout the solution process, ensuring that each step is logically sound and leads us closer to the desired outcome. Think of it like constructing a building: a strong foundation ensures the stability and integrity of the entire structure. Similarly, a firm grasp of algebraic fundamentals empowers us to confidently navigate complex equations and arrive at accurate solutions. This foundational knowledge not only helps us solve equations but also provides a deeper understanding of the relationships between variables and the mathematical principles that govern them.
2. Simplifying Both Sides of the Equation
The first step in solving our equation, 78x - 2x + 15y - x + 3x = 360 + 29 - 5 + 6, is to simplify both sides independently. This means combining like terms on the left side and performing the arithmetic operations on the right side. Let's start with the left side. We have several terms involving x: 78x, -2x, -x, and 3x. Combining these, we get 78x - 2x - x + 3x = (78 - 2 - 1 + 3)x = 78x. So, the left side simplifies to 78x + 15y. Now, let's tackle the right side: 360 + 29 - 5 + 6. Performing the addition and subtraction, we get 360 + 29 - 5 + 6 = 390. Therefore, the simplified equation becomes 78x + 15y = 390. Simplifying each side of the equation is a crucial step because it reduces the complexity and makes the equation easier to work with. It's like decluttering a room before you start organizing it – you need to get rid of the unnecessary distractions before you can focus on the core elements. In this case, simplifying allows us to clearly see the relationship between the variables and the constant terms. This clarity is essential for making informed decisions about the next steps in the solution process. By simplifying, we've transformed a complex-looking equation into a more manageable form, setting the stage for further manipulation and ultimately, finding the solution.
3. Further Simplification by Factoring
Looking at our simplified equation, 78x + 15y = 390, we can notice a common factor among the coefficients. This is a fantastic opportunity to further simplify the equation. The greatest common divisor (GCD) of 78, 15, and 390 is 3. We can factor out 3 from each term: 3(26x + 5y) = 3(130). Now, we can divide both sides of the equation by 3 to get 26x + 5y = 130. This step significantly reduces the magnitude of the numbers involved, making the equation easier to handle in subsequent steps. Factoring is a powerful technique in algebra because it allows us to express equations in a more concise and manageable form. It's like finding the common thread that runs through a complex pattern – by identifying and extracting the common factor, we reveal the underlying structure of the equation. In this case, factoring out 3 not only simplifies the coefficients but also highlights a potential relationship between the variables x and y. This simplification can be particularly helpful when we're trying to solve for one variable in terms of the other or when we're dealing with systems of equations. By reducing the equation to its simplest form, we make it easier to analyze, manipulate, and ultimately, solve. This step underscores the importance of looking for patterns and common factors in algebraic expressions, as it can often lead to significant simplifications and make the problem-solving process more efficient.
4. Expressing One Variable in Terms of the Other
At this point, we have the equation 26x + 5y = 130. Since we have two variables and only one equation, we cannot find unique numerical solutions for x and y. Instead, we can express one variable in terms of the other. This means isolating one variable on one side of the equation and expressing it as a function of the other variable. Let's solve for y. Subtract 26x from both sides: 5y = 130 - 26x. Now, divide both sides by 5: y = (130 - 26x) / 5. Simplifying further, we get y = 26 - (26/5)x. This equation tells us how y varies as x changes. For any value of x, we can plug it into this equation to find the corresponding value of y. Expressing one variable in terms of another is a fundamental technique in algebra, particularly when dealing with equations that have multiple solutions or when working with systems of equations. It allows us to understand the relationship between the variables and how they influence each other. In this case, we've established a clear relationship between x and y: y is a linear function of x. This means that as x increases, y decreases (since the coefficient of x is negative). This understanding can be incredibly valuable in various applications, such as modeling real-world phenomena or analyzing data. By expressing y in terms of x, we've gained a deeper insight into the equation and its possible solutions. This step highlights the power of algebraic manipulation in revealing the underlying structure and relationships within mathematical expressions.
5. Understanding Infinite Solutions
Because we have one equation and two unknowns (x and y), there are infinitely many solutions. Each pair of x and y values that satisfies the equation y = 26 - (26/5)x is a solution. For example, if we let x = 0, then y = 26. If we let x = 5, then y = 0. If we let x = 10, then y = -26. We can continue plugging in different values for x and find corresponding values for y that satisfy the equation. This is a key concept in algebra: when you have fewer equations than variables, you typically have infinitely many solutions. These solutions can be represented graphically as a line on a coordinate plane. Each point on the line corresponds to a pair of x and y values that satisfy the equation. Understanding the concept of infinite solutions is crucial for interpreting the results of algebraic manipulations. It's important to recognize that not all equations have a single, unique solution. In many real-world scenarios, we encounter situations where there are multiple possible outcomes, and algebraic equations can help us model and understand these scenarios. The fact that we have infinitely many solutions in this case doesn't mean that the equation is meaningless. On the contrary, it provides us with a powerful tool for exploring the relationship between x and y and for identifying specific solutions that might be relevant to a particular context. By recognizing the possibility of infinite solutions, we broaden our understanding of algebraic problem-solving and gain a deeper appreciation for the versatility of mathematical tools.
6. Practical Applications of Solving Equations
Solving equations like 78x - 2x + 15y - x + 3x = 360 + 29 - 5 + 6 isn't just an academic exercise; it has numerous practical applications in various fields. From physics and engineering to economics and computer science, the ability to manipulate and solve equations is essential for modeling real-world problems and finding solutions. For example, in physics, equations are used to describe the motion of objects, the flow of electricity, and the behavior of light. In engineering, equations are used to design structures, analyze circuits, and optimize processes. In economics, equations are used to model supply and demand, predict market trends, and analyze economic policies. In computer science, equations are used to develop algorithms, design databases, and model complex systems. The specific equation we solved today might not directly appear in these applications, but the underlying skills and techniques we used – simplifying expressions, combining like terms, isolating variables – are fundamental to solving a wide range of problems in these fields. Think of it like learning the alphabet: you might not use the letter "Q" every day, but knowing it is essential for reading and writing. Similarly, mastering algebraic techniques is essential for tackling complex problems in various disciplines. By developing a strong foundation in algebra, you're equipping yourself with a powerful toolkit that can be applied to a wide range of real-world challenges. This highlights the importance of not just memorizing formulas and procedures but also understanding the underlying concepts and principles. The ability to think critically and apply algebraic techniques creatively is what truly empowers you to solve problems effectively and make meaningful contributions in your chosen field.
7. Conclusion: Mastering Algebraic Techniques
Alright guys, we've walked through the entire process of solving the equation 78x - 2x + 15y - x + 3x = 360 + 29 - 5 + 6. We simplified, combined like terms, factored, and expressed one variable in terms of the other. We also learned that this equation has infinitely many solutions because we have two variables and only one equation. Remember, the key to mastering algebra is practice and understanding the underlying concepts. Don't be afraid to tackle challenging problems, and always break them down into smaller, more manageable steps. The skills you've learned today will serve you well in your future mathematical endeavors and in various real-world applications. From engineering and physics to economics and computer science, the ability to solve equations is a valuable asset. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. The more you engage with algebra, the more comfortable and confident you'll become. And remember, even the most complex problems can be solved with a systematic approach and a solid understanding of the fundamentals. This journey through algebraic problem-solving is not just about finding answers; it's about developing critical thinking skills, enhancing analytical abilities, and fostering a deeper appreciation for the power and elegance of mathematics. So, embrace the challenge, persevere through the difficulties, and celebrate your successes along the way. With dedication and consistent effort, you'll unlock the secrets of algebra and open doors to a world of possibilities.
