Solving 5(x+2)+4=8x-1-3(x-5) A Step-by-Step Guide

Algebraic equations are fundamental to mathematics and appear in various real-world applications. The ability to solve these equations is a crucial skill for anyone studying mathematics, science, or engineering. In this comprehensive guide, we will dissect the equation 5(x+2)+4=8x-1-3(x-5), providing a step-by-step solution and explaining the underlying principles. This equation is a linear equation, which means the highest power of the variable 'x' is 1. Solving linear equations involves isolating the variable on one side of the equation to determine its value. Before diving into the solution, it's essential to understand the properties of equality that allow us to manipulate equations without changing their solutions. These properties include the addition, subtraction, multiplication, and division properties of equality. These properties state that adding, subtracting, multiplying, or dividing both sides of an equation by the same non-zero number maintains the equality. Our goal is to use these properties strategically to simplify the equation and isolate the variable 'x'. Let's embark on this journey of solving the equation, breaking down each step to ensure a clear understanding of the process. By the end of this guide, you will not only be able to solve this specific equation but also have a solid foundation for tackling similar algebraic problems. The beauty of algebra lies in its systematic approach, and mastering these techniques opens doors to more advanced mathematical concepts. Remember, practice is key to proficiency in mathematics, so work through various examples to solidify your understanding. This comprehensive guide aims to provide the necessary tools and knowledge to confidently solve linear equations and appreciate their role in mathematics and beyond. As we progress, we will emphasize the importance of accuracy and attention to detail, which are crucial for success in solving equations. So, let's begin our exploration of solving 5(x+2)+4=8x-1-3(x-5) and unlock the power of algebraic manipulation.

1. Expanding the Expressions

The first step in solving the equation 5(x+2)+4=8x-1-3(x-5) is to expand the expressions within the parentheses. This involves applying the distributive property, which states that a(b+c) = ab + ac. By expanding, we eliminate the parentheses and simplify the equation. Starting with the left side of the equation, we have 5(x+2). Applying the distributive property, we multiply 5 by both x and 2: 5 * x = 5x and 5 * 2 = 10. Therefore, 5(x+2) expands to 5x + 10. Now, let's look at the right side of the equation, where we have -3(x-5). Again, we apply the distributive property, multiplying -3 by both x and -5: -3 * x = -3x and -3 * -5 = 15. Thus, -3(x-5) expands to -3x + 15. It's crucial to pay attention to the signs when applying the distributive property, as a negative multiplied by a negative results in a positive. Once we have expanded the expressions, the equation becomes 5x + 10 + 4 = 8x - 1 - 3x + 15. This simplified form is easier to work with because it removes the parentheses, allowing us to combine like terms. This step is a fundamental algebraic technique, and mastering it is crucial for solving various types of equations. Expanding the expressions correctly ensures that we maintain the equality of the equation while simplifying it. It's like rearranging the pieces of a puzzle to make it easier to solve. The next step involves combining like terms, which further simplifies the equation and brings us closer to isolating the variable x. So, by expanding the expressions, we've laid the groundwork for a successful solution. Remember to always double-check your work when applying the distributive property to avoid errors that can propagate through the rest of the solution. With the expressions expanded, we are now ready to proceed to the next step: combining like terms.

2. Combining Like Terms

After expanding the expressions in the equation 5(x+2)+4=8x-1-3(x-5), we arrived at 5x + 10 + 4 = 8x - 1 - 3x + 15. The next logical step is to combine like terms on both sides of the equation. Like terms are terms that contain the same variable raised to the same power, or constants. Combining like terms simplifies the equation and makes it easier to isolate the variable. On the left side of the equation, we have the constant terms 10 and 4. Adding these together gives us 10 + 4 = 14. So, the left side of the equation simplifies to 5x + 14. On the right side of the equation, we have two terms with the variable x: 8x and -3x. Combining these gives us 8x - 3x = 5x. We also have the constant terms -1 and 15. Adding these together gives us -1 + 15 = 14. Therefore, the right side of the equation simplifies to 5x + 14. Now, our equation looks significantly simpler: 5x + 14 = 5x + 14. This simplified form reveals an interesting characteristic of the equation. Combining like terms is a crucial algebraic technique because it reduces the complexity of the equation and brings us closer to isolating the variable. It's like organizing your tools before starting a project, making the task more manageable. By combining like terms, we ensure that we are working with the simplest possible form of the equation, minimizing the chances of making errors. This step also highlights the importance of carefully tracking the signs of the terms, as a mistake in addition or subtraction can lead to an incorrect solution. Once like terms are combined, we can better assess the nature of the equation and determine the next steps required to solve it. In this case, the simplified equation 5x + 14 = 5x + 14 suggests a special type of solution, which we will explore in the following steps. So, by mastering the technique of combining like terms, we are well-equipped to tackle a wide range of algebraic equations. The next step will reveal the intriguing nature of this particular equation and its solution. We have simplified the equation significantly, and now we are ready to uncover its unique solution.

3. Isolating the Variable

After combining like terms, the equation 5(x+2)+4=8x-1-3(x-5) simplified to 5x + 14 = 5x + 14. The next step in solving an equation is typically to isolate the variable, which means getting the variable term by itself on one side of the equation. To do this, we can use inverse operations to eliminate terms. In this case, we can subtract 5x from both sides of the equation. Subtracting 5x from the left side gives us 5x + 14 - 5x = 14. Subtracting 5x from the right side gives us 5x + 14 - 5x = 14. After performing this operation, the equation becomes 14 = 14. Notice that the variable x has completely disappeared from the equation. This indicates that the equation is either an identity or a contradiction. An identity is an equation that is true for all values of the variable, while a contradiction is an equation that is never true, regardless of the value of the variable. In this case, 14 = 14 is a true statement, which means the original equation is an identity. This implies that any real number substituted for x will satisfy the equation. When the variable disappears during the process of solving an equation, it's crucial to examine the resulting statement carefully. If the statement is true, as in this case, the equation is an identity. If the statement is false, the equation is a contradiction and has no solution. The process of isolating the variable is a fundamental technique in solving equations, but it's also important to interpret the results correctly. In some cases, isolating the variable may not lead to a unique solution, as we have seen here. This step highlights the importance of understanding the different types of equations and their solutions. So, while isolating the variable is a crucial skill, interpreting the outcome is equally important. In this particular case, the disappearance of the variable has revealed that the equation is an identity, which has significant implications for its solution. We have now reached a point where we can definitively state the solution to the equation. Let's proceed to the final step to summarize our findings and provide the complete solution.

4. Determining the Solution

Having simplified the equation 5(x+2)+4=8x-1-3(x-5) to 14 = 14, we have arrived at the crucial step of determining the solution. As we observed in the previous step, the variable x disappeared during the simplification process, leaving us with a true statement. This indicates that the original equation is an identity. An identity is an equation that holds true for all values of the variable. In other words, no matter what real number we substitute for x, the equation will always be satisfied. This is because the two sides of the equation are essentially the same expression, just written in different forms. Therefore, the solution to the equation is all real numbers. We can express this solution in several ways. One way is to write