Solve 3x + 4(x + 6) = 45: Step-by-Step Solution

Hey everyone! Today, we're going to dive into solving a simple algebraic equation. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you can follow along easily. The equation we're tackling is 3x + 4(x + 6) = 45. We'll go through the process of finding the value of 'x' that makes this equation true. So, grab your pencils and paper, and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation means. In simple terms, an equation is a mathematical statement that shows two expressions are equal. Our equation, 3x + 4(x + 6) = 45, has two sides: the left-hand side (LHS) which is 3x + 4(x + 6), and the right-hand side (RHS) which is 45. The goal is to find the value of 'x' that makes the LHS equal to the RHS. The variable 'x' represents an unknown number, and our job is to figure out what that number is. We'll use basic algebraic principles to isolate 'x' on one side of the equation. This involves performing operations (like addition, subtraction, multiplication, and division) on both sides of the equation to maintain the equality. Remember, whatever we do to one side, we must do to the other! This keeps the equation balanced. Now, let's move on to the actual solving process.

Step 1: Distribute the 4

The first step in solving our equation, 3x + 4(x + 6) = 45, involves dealing with the parentheses. We have a 4 multiplied by the expression (x + 6). To simplify this, we need to distribute the 4 across both terms inside the parentheses. This means we multiply 4 by both x and 6. So, 4 * x becomes 4x, and 4 * 6 becomes 24. Now, our equation looks like this: 3x + 4x + 24 = 45. By distributing, we've eliminated the parentheses, making the equation easier to work with. This step is crucial because it allows us to combine like terms in the next step. Remember, the distributive property is a fundamental concept in algebra, and it's essential for simplifying expressions and solving equations. So, make sure you're comfortable with this step before moving on. This distribution allows us to rewrite the equation in a more manageable form, paving the way for isolating the variable 'x'.

Step 2: Combine Like Terms

Now that we've distributed the 4, our equation is 3x + 4x + 24 = 45. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with 'x': 3x and 4x. To combine them, we simply add their coefficients (the numbers in front of the 'x'). So, 3x + 4x equals 7x. Our equation now becomes 7x + 24 = 45. By combining like terms, we've simplified the equation further, making it even easier to solve for 'x'. This step helps to consolidate the 'x' terms, bringing us closer to isolating 'x' on one side of the equation. Remember, only terms with the same variable and power can be combined. This principle of combining like terms is a cornerstone of algebraic manipulation and is crucial for solving a wide variety of equations.

Step 3: Subtract 24 from Both Sides

We're making great progress! Our equation is currently 7x + 24 = 45. To isolate 'x', we need to get rid of the + 24 on the left-hand side. We can do this by performing the inverse operation, which is subtraction. We subtract 24 from both sides of the equation. This is crucial to maintain the balance of the equation – whatever we do to one side, we must do to the other. Subtracting 24 from the left side cancels out the + 24, leaving us with just 7x. Subtracting 24 from the right side (45 - 24) gives us 21. So, our equation now looks like this: 7x = 21. We're one step closer to finding the value of 'x'! Subtracting a constant from both sides is a standard technique in solving equations and helps us to progressively isolate the variable we're trying to find.

Step 4: Divide Both Sides by 7

We're almost there! Our equation is now 7x = 21. To finally isolate 'x', we need to get rid of the 7 that's multiplying it. We do this by performing the inverse operation, which is division. We divide both sides of the equation by 7. Dividing the left side (7x) by 7 leaves us with just 'x'. Dividing the right side (21) by 7 gives us 3. So, our equation now looks like this: x = 3. We've found our solution! Dividing both sides by the coefficient of 'x' is the final step in isolating the variable and determining its value. This step effectively undoes the multiplication, leaving us with the value of 'x'. Congratulations, we've successfully solved for 'x'!

Solution: x = 3

So, after following all the steps, we've found that the solution to the equation 3x + 4(x + 6) = 45 is x = 3. This means that if we substitute 3 for 'x' in the original equation, both sides will be equal. Let's quickly check our answer to make sure it's correct. Substituting x = 3 into the original equation, we get: 3(3) + 4(3 + 6) = 45. Simplifying, we have: 9 + 4(9) = 45, which becomes 9 + 36 = 45. And indeed, 45 = 45, so our solution is correct! We've successfully solved the equation and verified our answer. Remember, it's always a good idea to check your solution to ensure accuracy. In the original question options, A. x = 3 is our answer. Great job, everyone!

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