Solving $3-x=-9$ A Step-by-Step Guide

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In this article, we will explore how to solve the equation $3-x=-9$. This is a fundamental algebraic problem that involves isolating the variable x to find its value. Understanding how to solve such equations is crucial for more advanced mathematical concepts. We will walk through the steps in a clear and detailed manner, ensuring that you grasp the underlying principles. This article aims to provide a comprehensive guide, making the process understandable for anyone, regardless of their prior mathematical experience. By the end of this guide, you will be able to confidently solve similar equations and understand the logic behind each step.

Understanding the Basics of Algebraic Equations

Before diving into the solution, it's important to grasp the basics of algebraic equations. An equation is a statement that two expressions are equal. In our case, the equation $3-x=-9$ states that the expression $3-x$ has the same value as $-9$. The goal of solving an equation is to find the value of the variable (in this case, x) that makes the equation true. This involves performing operations on both sides of the equation to isolate the variable. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance. This principle is the cornerstone of solving algebraic equations. The variable x represents an unknown quantity, and our objective is to determine what that quantity is. By understanding these fundamental concepts, we set the stage for a clear and effective solution process.

Step-by-Step Solution of the Equation $3-x=-9$

To solve the equation $3-x=-9$, we need to isolate the variable x. Here’s a step-by-step guide:

Step 1: Isolate the Term with x

Our first goal is to get the term containing x by itself on one side of the equation. In this case, we want to isolate $-x$. To do this, we need to eliminate the $3$ on the left side. We can achieve this by subtracting $3$ from both sides of the equation. This maintains the balance of the equation, ensuring that both sides remain equal. Subtracting $3$ from both sides gives us:

$3 - x - 3 = -9 - 3$

Simplifying this, we get:

$-x = -12$

Step 2: Solve for x

Now we have $-x = -12$. However, we want to find the value of x, not $-x$. To do this, we need to eliminate the negative sign in front of x. We can accomplish this by multiplying both sides of the equation by $-1$. This is because multiplying a negative number by $-1$ results in a positive number. So, multiplying both sides by $-1$, we have:

$-1 * (-x) = -1 * (-12)$

This simplifies to:

$x = 12$

Step 3: Verification

It's always a good practice to verify our solution. To verify that $x = 12$ is the correct solution, we substitute $12$ for x in the original equation and check if both sides of the equation are equal.

Original equation: $3 - x = -9$

Substitute $x = 12$:

$3 - 12 = -9$

Simplify:

$-9 = -9$

Since both sides of the equation are equal, our solution $x = 12$ is correct.

Detailed Explanation of Each Step

Step 1: Isolating the Term with x

The initial equation we have is $3-x=-9$. The first step towards solving for x involves isolating the term that contains x, which in this case is $-x$. To achieve this, we need to eliminate the $3$ from the left side of the equation. The operation that undoes addition is subtraction, so we subtract $3$ from both sides. This is a crucial step because it maintains the balance of the equation. If we subtract $3$ from only one side, the equation would no longer be true. By subtracting from both sides, we ensure that the equality remains valid.

Subtracting $3$ from both sides, we get:

$3 - x - 3 = -9 - 3$

On the left side, $3 - 3$ cancels out, leaving us with $-x$. On the right side, $-9 - 3$ simplifies to $-12$. Thus, the equation becomes:

$-x = -12$

This step is critical because it brings us closer to isolating x and determining its value. By carefully applying the subtraction property of equality, we've simplified the equation while maintaining its integrity.

Step 2: Solving for x

After isolating $-x$, we have the equation $-x = -12$. However, our ultimate goal is to find the value of x, not $-x$. The negative sign in front of x indicates that we have the opposite of x. To find x itself, we need to eliminate this negative sign. The mathematical operation that undoes negation is multiplication by $-1$. Therefore, we multiply both sides of the equation by $-1$.

Multiplying both sides by $-1$ gives us:

$-1 * (-x) = -1 * (-12)$

On the left side, $-1 * (-x)$ simplifies to x, because the product of two negative numbers is a positive number. On the right side, $-1 * (-12)$ simplifies to $12$ for the same reason. Thus, the equation becomes:

$x = 12$

This step is the culmination of our efforts to isolate x. By applying the multiplication property of equality, we've successfully solved for x, finding that its value is $12$. This is the solution to the equation $3-x=-9$.

Step 3: Verification of the Solution

Verification is an essential step in solving any equation. It ensures that our solution is correct and that no errors were made during the solving process. To verify the solution $x = 12$, we substitute this value back into the original equation $3 - x = -9$ and check if both sides of the equation are equal.

Substituting $x = 12$ into the original equation, we get:

$3 - 12 = -9$

Now, we simplify the left side of the equation:

$3 - 12 = -9$

Comparing both sides, we see that:

$-9 = -9$

Since both sides of the equation are equal, our solution $x = 12$ satisfies the original equation. This confirms that our solution is correct. Verification is a crucial step because it can help identify errors in the solving process, such as incorrect application of operations or arithmetic mistakes. By verifying the solution, we gain confidence in our answer and ensure the accuracy of our work.

Common Mistakes to Avoid

When solving equations like $3-x=-9$, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy. Here are some common mistakes to watch out for:

1. Incorrectly Applying Operations

One of the most common errors is not performing the same operation on both sides of the equation. Remember, to maintain the equality, any operation you perform on one side must also be performed on the other. For instance, in our equation, if you subtract $3$ only from the left side but not the right side, you will end up with an incorrect result. Always ensure that you apply the same operation to both sides to maintain the balance of the equation.

2. Sign Errors

Sign errors are particularly common when dealing with negative numbers. For example, when isolating $-x$, it's crucial to remember that you are solving for x, not $-x$. Multiplying or dividing by $-1$ is necessary to eliminate the negative sign. A mistake in handling negative signs can lead to an incorrect solution. Pay close attention to the signs of the numbers and variables throughout the solving process.

3. Misunderstanding Order of Operations

Understanding the order of operations (PEMDAS/BODMAS) is vital in solving equations. However, when solving for a variable, you often need to reverse the order of operations. For example, in $3-x=-9$, you first subtract $3$ from both sides before dealing with the negative sign on x. Misapplying the order of operations can lead to an incorrect solution.

4. Forgetting to Verify the Solution

Failing to verify the solution is a common oversight. Verification is a crucial step that confirms the accuracy of your solution. By substituting the solution back into the original equation, you can check if both sides of the equation are equal. If they are not, there's an error in your solving process, and you need to revisit your steps.

5. Difficulty with Integer Arithmetic

Working with integers, especially negative numbers, can be challenging for some. Mistakes in addition, subtraction, multiplication, or division of integers can lead to incorrect solutions. Ensure you have a solid understanding of integer arithmetic to avoid these errors. Practice with integer operations can significantly improve your accuracy.

By being mindful of these common mistakes, you can significantly improve your accuracy in solving algebraic equations. Always double-check your work, pay attention to details, and verify your solutions to minimize errors.

Alternative Methods to Solve the Equation

While the step-by-step method we discussed is a straightforward way to solve the equation $3-x=-9$, there are alternative approaches that can be used. Exploring these methods can provide a deeper understanding of algebraic problem-solving and offer different perspectives on the same problem.

Method 1: Adding x to Both Sides

Instead of subtracting $3$ from both sides as our first step, we can add x to both sides of the equation. This approach can sometimes simplify the process by immediately moving the x term to the right side.

Starting with the original equation:

$3 - x = -9$

Add x to both sides:

$3 - x + x = -9 + x$

Simplify:

$3 = -9 + x$

Now, to isolate x, add $9$ to both sides:

$3 + 9 = -9 + x + 9$

Simplify:

$12 = x$

Thus, we find that $x = 12$, which is the same solution we obtained using the step-by-step method.

Method 2: Rearranging the Equation

Another method involves rearranging the equation to a more familiar form. We can rewrite $3-x=-9$ as $-x+3=-9$. This might make it easier for some to visualize the next steps.

Starting with the rearranged equation:

$-x + 3 = -9$

Subtract $3$ from both sides:

$-x + 3 - 3 = -9 - 3$

Simplify:

$-x = -12$

Now, multiply both sides by $-1$ to solve for x:

$-1 * (-x) = -1 * (-12)$

Simplify:

$x = 12$

Again, we arrive at the same solution, $x = 12$.

These alternative methods demonstrate that there isn't always a single way to solve an equation. Depending on your understanding and preference, you can choose the method that feels most intuitive to you. The key is to maintain the balance of the equation by performing the same operations on both sides and to follow the rules of algebra correctly.

Practice Problems

To solidify your understanding of solving equations like $3-x=-9$, it's essential to practice with similar problems. Here are some practice problems that you can try:

1. Solve: $5 - x = -7$
2. Solve: $2 - x = -4$
3. Solve: $1 - x = -10$
4. Solve: $4 - x = -3$
5. Solve: $6 - x = -6$

For each of these problems, follow the steps we discussed earlier:

1. Isolate the term with x.
2. Solve for x.
3. Verify your solution by substituting it back into the original equation.

Solving these practice problems will help you become more comfortable and confident in solving algebraic equations. It's also a good idea to review your work and identify any mistakes you made. Understanding where you went wrong is a crucial part of the learning process.

Conclusion

In this article, we have thoroughly explored how to solve the equation $3-x=-9$. We started with the basics of algebraic equations, walked through the step-by-step solution process, and verified our answer. We also discussed common mistakes to avoid and presented alternative methods to solve the same equation. By understanding the fundamental principles and practicing consistently, you can confidently solve similar algebraic problems.

Remember, the key to mastering algebra is practice. Work through various problems, pay attention to detail, and always verify your solutions. With consistent effort, you will develop the skills needed to tackle more complex equations and mathematical concepts. We hope this guide has been helpful and has improved your understanding of solving algebraic equations.