Solving For X In The Equation 3(x-4)-5(x+2)=2-(x+24)

Introduction to Solving Linear Equations

In mathematics, solving for x in an equation is a fundamental skill that unlocks more advanced concepts. Equations are mathematical statements that assert the equality of two expressions. The variable 'x' represents an unknown value that we aim to find. Mastering the art of solving for x involves isolating the variable on one side of the equation by performing the same operations on both sides, maintaining the balance and equality. This process utilizes the properties of equality, which allow us to add, subtract, multiply, or divide both sides of an equation by the same value without changing its solution set. Linear equations, characterized by variables raised to the first power, are the building blocks of algebraic problem-solving. In the following sections, we will delve into a step-by-step approach to solve the equation 3(x-4) - 5(x+2) = 2 - (x+24), illustrating the application of algebraic principles to unravel the value of x. Understanding these techniques is not only crucial for academic pursuits but also for real-world applications where problem-solving often involves mathematical modeling and analysis. So, let's embark on this journey of algebraic exploration and discover the value of x in the given equation. By carefully applying the distributive property, combining like terms, and isolating the variable, we will demystify the process of solving for x and equip you with the skills to tackle similar challenges with confidence and precision. Remember, the key to success in mathematics lies in practice and a solid grasp of the underlying concepts. With dedication and a systematic approach, you can conquer the world of equations and unlock the power of algebraic thinking.

Understanding the Given Equation

Before diving into the solution, let's take a moment to understand the given equation: 3(x-4) - 5(x+2) = 2 - (x+24). This equation is a linear equation, which means the highest power of the variable 'x' is 1. Linear equations are the simplest type of algebraic equations, and they form the foundation for more complex mathematical concepts. Our goal is to isolate 'x' on one side of the equation to determine its value. The equation involves several operations, including multiplication, subtraction, and addition, and it also includes parentheses, which indicate the order of operations. To effectively solve this equation, we need to follow a systematic approach, applying the distributive property to eliminate parentheses, combining like terms to simplify the equation, and using inverse operations to isolate 'x'. The presence of parentheses in the equation signifies the importance of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We must first address the expressions within the parentheses before proceeding with other operations. This ensures that we maintain the mathematical integrity of the equation and arrive at the correct solution. Furthermore, the equation involves both positive and negative terms, which requires careful attention to signs when performing operations. A small error in sign manipulation can lead to an incorrect solution. Therefore, it is crucial to pay close attention to detail and double-check each step to ensure accuracy. By understanding the structure and components of the equation, we can approach the solving process with confidence and clarity, avoiding common pitfalls and arriving at the correct value of 'x'. Remember, a thorough understanding of the equation is the first step towards finding its solution. Let's proceed to the next step, where we will apply the distributive property to eliminate the parentheses and further simplify the equation.

Step 1: Applying the Distributive Property

To begin solving the equation, we must first apply the distributive property. The distributive property states that a(b + c) = ab + ac. This property allows us to eliminate the parentheses in our equation, making it easier to work with. Looking at our equation, 3(x-4) - 5(x+2) = 2 - (x+24), we have two instances where we need to apply the distributive property: 3(x-4) and -5(x+2). For the first term, 3(x-4), we multiply 3 by both x and -4, resulting in 3x - 12. For the second term, -5(x+2), we multiply -5 by both x and 2, resulting in -5x - 10. It's crucial to pay attention to the negative sign in front of the 5, as it affects the signs of both terms inside the parentheses. Now, let's rewrite the equation with the distributive property applied: 3x - 12 - 5x - 10 = 2 - (x+24). We still have one set of parentheses left: -(x+24). This can be treated as -1(x+24), so we multiply -1 by both x and 24, resulting in -x - 24. Now our equation looks like this: 3x - 12 - 5x - 10 = 2 - x - 24. By applying the distributive property, we have successfully eliminated all parentheses from the equation. This is a significant step forward, as it allows us to combine like terms and further simplify the equation. The distributive property is a fundamental concept in algebra, and mastering its application is essential for solving equations and simplifying expressions. It is a powerful tool that allows us to transform complex expressions into simpler, more manageable forms. With the parentheses gone, we can now move on to the next step, which involves combining like terms on both sides of the equation. This will further simplify the equation and bring us closer to isolating the variable 'x'.

Step 2: Combining Like Terms

After applying the distributive property, the next crucial step is combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 3x - 12 - 5x - 10 = 2 - x - 24, we can identify like terms on both sides of the equation. On the left side, we have 3x and -5x, which are like terms because they both contain the variable 'x' raised to the power of 1. We also have -12 and -10, which are constant terms and therefore also like terms. On the right side, we have 2 and -24, which are constant terms and like terms. To combine like terms, we simply add or subtract their coefficients. For the 'x' terms on the left side, we have 3x - 5x, which equals -2x. For the constant terms on the left side, we have -12 - 10, which equals -22. So, the left side of the equation simplifies to -2x - 22. On the right side, we have the constant terms 2 - 24, which equals -22. The right side also has the term -x, which we will keep as is for now. So, the right side of the equation is -x - 22. Now, our equation looks like this: -2x - 22 = -x - 22. By combining like terms, we have simplified the equation significantly. This makes it easier to isolate the variable 'x' and solve for its value. Combining like terms is a fundamental skill in algebra, and it is essential for solving equations efficiently. It allows us to reduce the complexity of an equation and make it more manageable. By identifying and combining like terms, we can streamline the solving process and minimize the chances of making errors. With the equation simplified, we are now ready to move on to the next step, which involves isolating the variable 'x' on one side of the equation. This will ultimately lead us to the solution for 'x'.

Step 3: Isolating the Variable x

Now that we have simplified the equation to -2x - 22 = -x - 22, the next step is isolating the variable x. This means we want to get all the 'x' terms on one side of the equation and all the constant terms on the other side. To do this, we can use the properties of equality, which allow us to add or subtract the same value from both sides of the equation without changing its solution. Let's start by adding 'x' to both sides of the equation. This will eliminate the '-x' term on the right side: -2x - 22 + x = -x - 22 + x. Simplifying this, we get -x - 22 = -22. Next, we want to get rid of the constant term on the left side, which is -22. To do this, we add 22 to both sides of the equation: -x - 22 + 22 = -22 + 22. Simplifying this, we get -x = 0. Finally, to solve for 'x', we need to get rid of the negative sign in front of the 'x'. We can do this by multiplying both sides of the equation by -1: -1 * (-x) = -1 * 0. This gives us x = 0. By isolating the variable 'x', we have successfully found its value. The process of isolating the variable involves using inverse operations to undo the operations that are being performed on the variable. In this case, we used addition and multiplication to isolate 'x'. Isolating the variable is a crucial step in solving equations, and it requires a solid understanding of the properties of equality. By carefully applying these properties, we can manipulate the equation to get the variable by itself on one side, allowing us to determine its value. With the value of 'x' now determined, we have solved the equation. However, it's always a good practice to check our solution to ensure that it is correct.

Step 4: Checking the Solution

After solving for x, it's crucial to check the solution to ensure accuracy. This involves substituting the value we found for x back into the original equation and verifying that both sides of the equation are equal. Our original equation was 3(x-4) - 5(x+2) = 2 - (x+24), and we found that x = 0. Let's substitute x = 0 into the equation: 3(0-4) - 5(0+2) = 2 - (0+24). Now, we simplify each side of the equation separately. On the left side, we have: 3(0-4) = 3(-4) = -12 and -5(0+2) = -5(2) = -10. So, the left side becomes -12 - 10 = -22. On the right side, we have: 2 - (0+24) = 2 - 24 = -22. Now, we compare both sides of the equation: -22 = -22. Since both sides are equal, our solution x = 0 is correct. Checking the solution is an essential step in the problem-solving process. It helps us identify any errors we may have made along the way and ensures that our answer is valid. By substituting the solution back into the original equation, we can verify that the equation holds true. This gives us confidence in our solution and confirms that we have successfully solved the problem. Furthermore, checking the solution can also help us develop a deeper understanding of the equation and the relationships between the variables. It reinforces the concept of equality and the importance of maintaining balance in the equation. In this case, our check confirmed that x = 0 is indeed the correct solution. This means that when x is equal to 0, the equation 3(x-4) - 5(x+2) = 2 - (x+24) is a true statement. With the solution verified, we can confidently conclude that we have successfully solved for x.

Conclusion: The Value of x

In conclusion, after a step-by-step process of applying the distributive property, combining like terms, isolating the variable, and checking the solution, we have successfully determined the value of x in the equation 3(x-4) - 5(x+2) = 2 - (x+24). The solution we found is x = 0. This means that when x is equal to 0, the equation holds true, and both sides of the equation are equal. Solving for x is a fundamental skill in algebra, and it is essential for understanding and solving more complex mathematical problems. The process involves using algebraic principles and properties to manipulate the equation and isolate the variable. By following a systematic approach, we can confidently solve for x in a wide range of equations. The distributive property allows us to eliminate parentheses, making the equation easier to work with. Combining like terms simplifies the equation by grouping similar terms together. Isolating the variable involves using inverse operations to get the variable by itself on one side of the equation. And checking the solution ensures that our answer is correct and that we have not made any errors along the way. The value of x represents the point at which the equation is balanced, where the expressions on both sides are equal. In this case, x = 0 is the solution that satisfies the equation. Understanding the process of solving for x not only enhances our mathematical skills but also develops our problem-solving abilities. It teaches us to think logically, follow a systematic approach, and pay attention to detail. These skills are valuable not only in mathematics but also in various aspects of life. So, mastering the art of solving for x is a worthwhile endeavor that will benefit us in numerous ways. With practice and a solid understanding of the underlying principles, we can confidently tackle equations and unlock the power of algebraic thinking.