Solving Equations: 3x + 2 = X + 4 Using Substitution

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Hey guys! Let's dive into solving the equation 3x + 2 = x + 4 using the substitution method. This is a fundamental concept in algebra, and understanding it will lay a solid foundation for more complex mathematical problems. We'll break it down step-by-step, making sure it's super clear and easy to follow. Ready? Let's get started!

Understanding the Substitution Method: The Core Idea

Alright, so what exactly is the substitution method? In simple terms, it's a way to solve equations by isolating a variable and then replacing it with its equivalent value in another equation. In this case, since we only have one equation, the goal is to isolate x on one side of the equation and then determine its value. The substitution method is particularly useful when dealing with systems of equations, but the core principle applies here: we manipulate the equation to express one variable in terms of others, and then use that expression to find the value of the variable. We can think of it as a mathematical puzzle where we rearrange the pieces until we find the solution. The key is to keep the equation balanced throughout the process. Whatever we do to one side, we must do to the other. This ensures that the equality remains true and we don’t inadvertently change the solution. It's like a seesaw; to keep it level, you must add or remove the same weight on both sides.

Let’s start with the given equation 3x + 2 = x + 4. Our aim is to isolate x. We can do this through a series of algebraic manipulations: adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression. Each step brings us closer to unraveling the mystery of what x really is. Remember that our primary objective is to get x alone on one side. This is achieved through systematic steps that don't violate the fundamental rules of mathematics. We are essentially unwrapping a mathematical package, layer by layer, until we reveal the unknown value inside.

So, why is this method called substitution? Because, once we know the value of x, we can substitute it back into the original equation (or any equivalent form) to verify our answer. This act of substitution is like double-checking our work. We're making sure that the value of x we found truly satisfies the initial conditions of the equation. This check provides an important safeguard against errors, ensuring we arrive at a correct solution. It's similar to proofreading a document. We substitute our solution into the original equation to ensure that it holds true.

Step-by-Step Solution: Unraveling 3x + 2 = x + 4

Let's get our hands dirty and solve this equation step by step, guys! We'll start with the equation: 3x + 2 = x + 4. Follow along closely, and you'll become a pro in no time! Here’s how we'll do it.

Step 1: Isolate the x terms

Our first goal is to get all the x terms on one side of the equation. To do this, let's subtract x from both sides. Remember, whatever we do to one side, we must do to the other to keep things balanced. So, we'll subtract x from both sides of the equation 3x + 2 = x + 4. This gives us:

  • 3x - x + 2 = x - x + 4

Simplifying this, we get:

  • 2x + 2 = 4

We successfully managed to move the x terms to one side. Great job!

Step 2: Isolate the constant terms

Now, let's move the constant terms (the numbers without x) to the other side. We'll subtract 2 from both sides of the equation 2x + 2 = 4. This gives us:

  • 2x + 2 - 2 = 4 - 2

Simplifying, we have:

  • 2x = 2

We're getting closer to isolating x! One more step and we are there.

Step 3: Solve for x

Finally, to solve for x, we need to get x completely by itself. We have 2x = 2. To isolate x, we'll divide both sides of the equation by 2:

  • (2x) / 2 = 2 / 2

This gives us:

  • x = 1

Ta-da! We've found the solution! The value of x is 1.

Verification: Checking Our Solution

It’s always a good idea to check your solution to make sure it's correct. We're going to substitute the value of x we found (which is 1) back into the original equation 3x + 2 = x + 4. This is our way of double-checking that our answer is right. Remember, this step is crucial because it ensures the equation is balanced and the equality holds true. Here’s what we do:

Substitute x = 1 into the equation: 3(1) + 2 = (1) + 4

Simplifying the left side, we get: 3 + 2 = 5

Simplifying the right side, we get: 1 + 4 = 5

So, we have: 5 = 5

Since both sides are equal, our solution x = 1 is correct! Awesome, right?

Conclusion: Mastering the Substitution Method

Congratulations, guys! You've successfully solved the equation 3x + 2 = x + 4 using the substitution method! We've seen how to isolate x step by step and then verify our answer. This process forms the basis for solving more complicated equations and systems of equations. Keep practicing, and you'll become a master in no time! Remember the key steps: isolate x terms, isolate constant terms, and solve for x. Always check your work by substitution to make sure your answer is correct. Understanding this approach is a valuable skill in algebra. The ability to manipulate equations and isolate variables is fundamental to solving problems in many different fields of mathematics, science, and even engineering.

In essence, we've navigated the algebraic terrain and found our treasure – the value of x. The substitution method is more than just a technique; it is a way of logical thinking. As you tackle more complex problems, you’ll find this method is an excellent building block to achieve greater things. With practice and understanding, you can apply these skills to solve a wide range of algebraic problems.

Keep exploring, keep practicing, and don't be afraid to ask questions. You've got this!