Solving For X In The Equation 2(x-3)+9=3(x+1)+x
In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a code, where we manipulate symbols and numbers to uncover the hidden value of an unknown variable. Today, we embark on a journey to solve a specific equation and reveal the value of the elusive 'x'. The equation that stands before us is: 2(x-3)+9=3(x+1)+x
To find the value of x, we need to carefully dissect the equation, applying the rules of algebra to isolate x on one side. This involves a series of steps, each a deliberate move in our quest to uncover the solution. Our adventure into the heart of this equation begins now.
Decoding the Equation: Step-by-Step
The first step in solving this equation is to simplify both sides by applying the distributive property. This property allows us to multiply a number outside the parentheses by each term inside. On the left side, we have 2(x-3), which expands to 2x - 23, or 2x - 6. Adding the +9 that was already there, the left side becomes 2x - 6 + 9.
Moving to the right side, we encounter 3(x+1), which expands to 3x + 31, or 3x + 3. We also have an additional +x term on this side. So, the right side of the equation becomes 3x + 3 + x. With the distributive property applied, our equation now looks like this:
2x - 6 + 9 = 3x + 3 + x
This is a crucial step, as it eliminates the parentheses and sets the stage for further simplification. We've essentially unwrapped the equation, making it easier to handle. Now, we move on to the next stage of our mathematical expedition.
Combining Like Terms: Bringing Order to the Equation
Now that we've applied the distributive property, it's time to combine like terms on both sides of the equation. Like terms are those that have the same variable raised to the same power. On the left side, we have -6 and +9, which are both constants. Combining them, -6 + 9 equals 3. So, the left side simplifies to 2x + 3.
On the right side, we have 3x and x, which are both terms with the variable x raised to the power of 1. Combining them, 3x + x equals 4x. The right side also has a constant term, +3. Thus, the right side simplifies to 4x + 3. Our equation now looks much cleaner:
2x + 3 = 4x + 3
This step is like tidying up a room, grouping similar items together. By combining like terms, we've streamlined the equation, making it more manageable and revealing its underlying structure. With the equation in this simplified form, we're ready to take the next step towards isolating x.
Isolating the Variable: The Quest for x
Our next goal is to isolate the variable x on one side of the equation. This means we want to get all the x terms on one side and all the constant terms on the other. To do this, we can use the properties of equality, which state that we can add or subtract the same value from both sides of an equation without changing its solution.
Let's start by subtracting 2x from both sides. This will eliminate the x term on the left side. Subtracting 2x from both sides of 2x + 3 = 4x + 3 gives us:
2x + 3 - 2x = 4x + 3 - 2x
Simplifying, we get:
3 = 2x + 3
Now, we need to get rid of the constant term (+3) on the right side. We can do this by subtracting 3 from both sides:
3 - 3 = 2x + 3 - 3
Simplifying again, we have:
0 = 2x
We're getting closer! We now have all the x terms on one side and all the constant terms on the other. The equation is becoming clearer, like a photograph developing in a darkroom. One final step remains to unveil the value of x.
Solving for x: The Grand Finale
We've reached the final step in our quest: solving for x. Our equation currently reads 0 = 2x. To isolate x completely, we need to get rid of the coefficient 2 that's multiplying it. We can do this by dividing both sides of the equation by 2. Dividing both sides of 0 = 2x by 2 gives us:
0 / 2 = 2x / 2
Simplifying, we get:
0 = x
Or, more conventionally:
x = 0
We've done it! We've successfully navigated the equation and discovered the value of x. It turns out that x is equal to 0. This is the solution to the equation 2(x-3)+9=3(x+1)+x. Our mathematical journey has reached its destination, and we can confidently state the answer.
Conclusion: The Value Revealed
In conclusion, by carefully applying the principles of algebra, we've solved the equation 2(x-3)+9=3(x+1)+x and found that the value of x is indeed 0. This process involved several key steps: distributing, combining like terms, isolating the variable, and finally, solving for x. Each step was crucial in unraveling the equation and revealing its hidden solution.
Therefore, the correct answer to the question "What is the value of x in the equation 2(x-3)+9=3(x+1)+x?" is:
C. x = 0
This exercise demonstrates the power of algebra in solving mathematical puzzles. By understanding the rules and applying them systematically, we can unlock the solutions to even the most complex equations. Mathematics is not just about numbers; it's about logic, problem-solving, and the thrill of discovery.
Answering Common Questions About Solving Equations
When tackling equations like the one we just solved, several questions often arise. Let's address some of these common queries to further enhance your understanding of equation-solving.
Why do we need to isolate the variable?
Isolating the variable is the core strategy in solving equations. The main goal is to determine the value of the unknown variable (in our case, 'x') that makes the equation true. By isolating the variable on one side of the equation, we effectively reveal its value. It's like peeling away layers of an onion, gradually exposing the core. Each step we take, whether it's adding, subtracting, multiplying, or dividing, is aimed at simplifying the equation and bringing the variable closer to being alone on one side.
Think of it like this: the equation is a balanced scale. The left side must equal the right side. To find the weight of 'x', we need to remove everything else from its side of the scale, one step at a time, while maintaining the balance. Once 'x' is alone, its weight (or value) is clear.
What is the distributive property and why is it important?
The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that a(b + c) = ab + ac. In simpler terms, it means we can multiply a number outside the parentheses by each term inside the parentheses. This property is crucial because it allows us to eliminate parentheses, which often stand in the way of simplifying an equation. In our example, we used the distributive property to expand 2(x - 3) and 3(x + 1).
Without the distributive property, we'd be stuck with the parentheses, making it difficult to combine like terms and isolate the variable. It's like having a set of building blocks stuck together; the distributive property is the tool that helps us separate them so we can work with them individually.
How do we combine like terms?
Combining like terms is another essential step in simplifying equations. Like terms are those that have the same variable raised to the same power (e.g., 2x and 3x) or are constants (e.g., 5 and -2). We can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables). For instance, 2x + 3x = 5x. Combining like terms helps to tidy up the equation, making it less cluttered and easier to solve. It's like organizing a messy room by grouping similar items together.
In our equation, we combined the 'x' terms on the right side (3x + x) and the constant terms on both sides (-6 + 9 and +3). This simplification is crucial because it reduces the number of terms in the equation, making it more manageable.
Why do we perform the same operation on both sides of the equation?
The principle of performing the same operation on both sides of the equation is based on the fundamental concept of equality. An equation is like a balanced scale; both sides must remain equal for the equation to be true. If we add, subtract, multiply, or divide only one side, we'll upset the balance. To maintain the equality, we must perform the same operation on both sides. This ensures that the equation remains balanced and the solution remains valid.
This principle is applied throughout the equation-solving process. When we subtract 2x from both sides or divide both sides by 2, we're adhering to this rule. It's like adding or removing the same weight from both sides of a scale to keep it balanced.
What if the equation has no solution or infinite solutions?
Not all equations have a single, unique solution. Some equations may have no solution, meaning there's no value of the variable that makes the equation true. This typically happens when the equation simplifies to a contradiction, such as 2 = 3. Other equations may have infinite solutions, meaning any value of the variable will satisfy the equation. This usually occurs when the equation simplifies to an identity, such as x = x or 5 = 5.
In these cases, the process of solving the equation will reveal the nature of the solution. If you arrive at a contradiction, there's no solution. If you arrive at an identity, there are infinite solutions. Understanding these possibilities is crucial for a complete understanding of equation-solving.
By addressing these common questions, we hope to provide a more comprehensive understanding of the equation-solving process. Mathematics is a journey of exploration, and each equation is a new puzzle to be solved. The more we understand the underlying principles, the better equipped we are to tackle any mathematical challenge.
