Solve 2x + 8 + 6x + 4 = 20: A Step-by-Step Guide

Hey guys! Let's dive into solving the equation 2x + 8 + 6x + 4 = 20. This is a classic algebra problem, and breaking it down step-by-step makes it super manageable. We'll cover everything from the basic principles to the nitty-gritty details, so you'll be solving similar problems like a pro in no time. So, grab your pencils, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into the specifics of this equation, it's essential to grasp the fundamental concepts of algebraic equations. An equation, at its core, is a statement asserting the equality of two expressions. These expressions can involve numbers, variables (like our 'x'), and mathematical operations. The goal when solving an equation is to isolate the variable—to get it all by itself on one side of the equals sign. This gives us the value of the variable that makes the equation true.

Think of an equation like a balancing scale. The equals sign (=) is the fulcrum, and the expressions on either side are the weights. To keep the scale balanced, whatever operation you perform on one side, you must perform on the other. This principle is crucial for maintaining the equation's integrity throughout the solving process. We maintain the balance by performing the same mathematical operations on both sides of the equation. For instance, if we subtract a number from the left side, we must subtract the same number from the right side to keep the equation balanced. Similarly, if we multiply the left side by a number, we must multiply the right side by the same number.

In our equation, 2x + 8 + 6x + 4 = 20, the variable is 'x'. Our mission is to find the value of 'x' that satisfies this equation. This involves a series of steps, each designed to simplify the equation while adhering to the fundamental principle of balance. To successfully solve algebraic equations, it's crucial to understand the properties of equality. These properties allow us to manipulate equations while preserving their balance. Some key properties include: The Addition Property of Equality, which states that adding the same number to both sides of an equation does not change its solution. The Subtraction Property of Equality, which states that subtracting the same number from both sides of an equation does not change its solution. The Multiplication Property of Equality, which states that multiplying both sides of an equation by the same nonzero number does not change its solution. The Division Property of Equality, which states that dividing both sides of an equation by the same nonzero number does not change its solution. By applying these properties, we can systematically isolate the variable and find its value. With a solid understanding of these basics, we can confidently tackle more complex equations. So, let's move on to the first step in solving our equation: combining like terms.

Step 1: Combining Like Terms in the Equation

The first step in simplifying the equation 2x + 8 + 6x + 4 = 20 is to combine like terms. Like terms are those that have the same variable raised to the same power (in this case, terms with 'x' and constant terms). This process makes the equation cleaner and easier to work with. It's like tidying up before you start a big project – it helps you see everything clearly. Combining like terms is a fundamental technique in algebra, and mastering it is crucial for solving equations efficiently. By combining like terms, we reduce the complexity of the equation, making it easier to isolate the variable and find its value. This step often involves addition or subtraction, depending on the signs of the terms. For instance, in our equation, we have two terms with the variable 'x': 2x and 6x. We also have two constant terms: 8 and 4. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables) or add the constants together.

Let’s start with the 'x' terms. We have 2x and 6x. When we add these together, we get 2x + 6x = 8x. It’s like saying you have 2 apples and then you get 6 more apples – now you have 8 apples. The 'x' is just a label, a placeholder for a number we don't know yet. Next, we'll combine the constant terms. We have 8 and 4. Adding these gives us 8 + 4 = 12. These are just regular numbers, so we can add them directly. Now, we can rewrite our original equation, replacing the separate 'x' terms and constants with their combined forms. So, 2x + 8 + 6x + 4 = 20 becomes 8x + 12 = 20. See how much simpler it looks? This is the power of combining like terms. It condenses the equation, making the next steps much clearer. By combining like terms, we've simplified the equation to a more manageable form. Now, we can proceed to the next step, which involves isolating the variable term. This is a critical step in solving the equation, as it brings us closer to finding the value of 'x'. In the following section, we'll discuss how to isolate the variable term by using inverse operations.

Step 2: Isolating the Variable Term

Now that we've combined like terms, our equation is 8x + 12 = 20. The next step is to isolate the variable term, which in this case is 8x. This means we need to get the 8x by itself on one side of the equation. To do this, we use inverse operations. Think of inverse operations as the opposites of mathematical operations. Addition and subtraction are inverses of each other, and multiplication and division are inverses of each other. The goal here is to undo the operations that are keeping the 8x from being alone. In our equation, we have 8x + 12. The + 12 is what's keeping the 8x company, so we need to undo the addition of 12. The inverse operation of addition is subtraction. So, to get rid of the + 12, we subtract 12 from both sides of the equation. Remember that golden rule of equations: whatever you do to one side, you must do to the other to keep the equation balanced. This is where the concept of maintaining balance in the equation comes into play. By subtracting 12 from both sides, we ensure that the equation remains balanced, and we move closer to isolating the variable term.

So, let’s subtract 12 from both sides: 8x + 12 - 12 = 20 - 12. On the left side, the + 12 and - 12 cancel each other out, leaving us with just 8x. On the right side, 20 - 12 = 8. So, our equation now looks like this: 8x = 8. We’ve successfully isolated the variable term! We're one step closer to finding the value of 'x'. This step is crucial because it simplifies the equation further, making it easier to solve for the variable. By isolating the variable term, we've set the stage for the final step: solving for 'x'. Now that the variable term is isolated, we can move on to the next step, which involves dividing both sides of the equation by the coefficient of 'x'. This will give us the value of 'x' and complete the solution.

Step 3: Solving for 'x'

We've reached the final step! Our equation is now 8x = 8. To solve for 'x', we need to get 'x' completely by itself. Currently, 'x' is being multiplied by 8. So, to undo this multiplication, we need to use the inverse operation: division. We'll divide both sides of the equation by 8. Again, we must do the same thing to both sides to keep the equation balanced. This is the final step in isolating the variable and finding its value. By dividing both sides by the coefficient of 'x', we effectively undo the multiplication and reveal the value of 'x'. This step is crucial for obtaining the solution to the equation.

So, let’s divide both sides by 8: (8x) / 8 = 8 / 8. On the left side, the 8 in the numerator and the 8 in the denominator cancel each other out, leaving us with just 'x'. On the right side, 8 / 8 = 1. Therefore, our solution is x = 1. We've done it! We've successfully solved the equation. This final step is the culmination of all the previous steps, and it provides the value of the variable that satisfies the equation. By understanding and applying the principles of inverse operations and maintaining balance in the equation, we can confidently solve for 'x'. Now that we have the solution, it's a good idea to verify it by substituting it back into the original equation.

Verification: Plugging the Solution Back In

To make sure our solution x = 1 is correct, we can plug it back into the original equation: 2x + 8 + 6x + 4 = 20. This is a crucial step in the problem-solving process, as it confirms the accuracy of our solution. By substituting the value of 'x' back into the original equation, we can verify whether it satisfies the equation or not. If the equation holds true after substitution, it confirms that our solution is correct. If not, it indicates that there might be an error in our calculations, and we need to retrace our steps to identify and correct the mistake. This practice helps build confidence in our problem-solving abilities and ensures that we arrive at the correct answer.

Let’s substitute x = 1 into the equation: 2(1) + 8 + 6(1) + 4 = 20. Now, let's simplify: 2 + 8 + 6 + 4 = 20. Adding the numbers on the left side, we get 20 = 20. This is a true statement! This confirms that our solution x = 1 is indeed correct. Verification is a great habit to get into when solving equations. It gives you peace of mind knowing you’ve got the right answer. By plugging the solution back into the original equation, we can catch any errors that might have occurred during the solving process. This not only ensures the accuracy of our answer but also helps us develop a deeper understanding of the equation and the solution.

Conclusion: Mastering Algebraic Equations

So, guys, we've successfully solved the equation 2x + 8 + 6x + 4 = 20, and found that x = 1. We walked through combining like terms, isolating the variable term, solving for 'x', and even verified our solution. Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts. By understanding the basic principles and practicing regularly, you can become proficient in solving a wide range of equations.

Remember, the key to success in algebra is practice and patience. Don't be discouraged if you encounter challenges along the way. Each problem is an opportunity to learn and grow. By breaking down complex equations into simpler steps, you can tackle them with confidence. So, keep practicing, and you'll become a math whiz in no time! You've got this!