Solve For { X $}$ In The Equation:${ -9(-3x - 1) - 3x = -39 }$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, step by step, to help readers understand the process and build their confidence in tackling similar problems.

The Equation

The equation we will be solving is:

$$-9(-3x - 1) - 3x = -39$$

This equation appears complex, but with a systematic approach, we can break it down and solve for the variable $x$.

Step 1: Distribute the Negative Sign

The first step in solving this equation is to distribute the negative sign to the terms inside the parentheses. This will help us simplify the equation and make it easier to work with.

$$-9(-3x - 1) = 27x + 9$$

Now, the equation becomes:

$$27x + 9 - 3x = -39$$

Step 2: Combine Like Terms

The next step is to combine like terms, which involves adding or subtracting terms that have the same variable and coefficient.

$$27x - 3x = 24x$$

So, the equation becomes:

$$24x + 9 = -39$$

Step 3: Isolate the Variable

Now, we need to isolate the variable $x$ by getting rid of the constant term on the left-hand side of the equation. We can do this by subtracting 9 from both sides of the equation.

$$24x = -39 - 9$$

$$24x = -48$$

Step 4: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by 24.

$$x = \frac{-48}{24}$$

$$x = -2$$

Conclusion

Solving linear equations requires a systematic approach, and breaking down the problem into smaller steps can make it more manageable. By following the steps outlined in this article, we were able to solve the equation $-9(-3x - 1) - 3x = -39$ and find the value of $x$. With practice and patience, readers can develop their skills in solving linear equations and become more confident in tackling complex mathematical problems.

Tips and Tricks

• Always start by simplifying the equation and getting rid of any parentheses or brackets.
• Combine like terms to make the equation easier to work with.
• Isolate the variable by getting rid of the constant term on the left-hand side of the equation.
• Finally, solve for the variable by dividing both sides of the equation by the coefficient.

Real-World Applications

Solving linear equations has numerous real-world applications, including:

• Physics: Solving linear equations is essential in physics, where equations are used to describe the motion of objects and the behavior of physical systems.
• Engineering: Linear equations are used in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Solving linear equations is used in economics to model and analyze economic systems, including supply and demand curves.

Common Mistakes

• Failing to simplify the equation before solving it.
• Not combining like terms.
• Not isolating the variable.
• Not checking the solution by plugging it back into the original equation.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving a linear equation. However, we understand that sometimes, it's not just about following a set of steps, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we'll tackle some of the most frequently asked questions about solving linear equations, providing clarity and insight to help readers better grasp the subject.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear, look for the highest power of the variable(s). If it's 1, then the equation is linear. If it's greater than 1, then the equation is not linear.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation has a highest power of 1, while a quadratic equation has a highest power of 2. For example, the equation 2x + 3 = 5 is linear, while the equation x^2 + 2x + 1 = 0 is quadratic.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, follow these steps:

1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
2. Simplify the equation by combining like terms.
3. Solve for the variable using the usual methods.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the underlying concepts and be able to solve equations by hand. This will help you develop problem-solving skills and avoid relying solely on technology.

Q: How do I check my solution to a linear equation?

A: To check your solution, plug the value back into the original equation and simplify. If the equation holds true, then your solution is correct.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Failing to simplify the equation before solving it.
• Not combining like terms.
• Not isolating the variable.
• Not checking the solution by plugging it back into the original equation.

Q: Can I use algebraic methods to solve linear equations?

A: Yes, you can use algebraic methods to solve linear equations. Some common methods include:

• Adding or subtracting the same value to both sides of the equation.
• Multiplying or dividing both sides of the equation by the same value.
• Using inverse operations to isolate the variable.

Conclusion

Solving linear equations is a fundamental skill that requires practice and patience. By understanding the underlying concepts and addressing common questions and concerns, readers can develop their skills in solving linear equations and become more confident in tackling complex mathematical problems. With practice and persistence, readers can master the art of solving linear equations and apply it to real-world problems.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• IXL: Linear Equations

Final Tips

• Practice solving linear equations regularly to develop your skills.
• Use algebraic methods to solve linear equations.
• Check your solution by plugging it back into the original equation.
• Avoid common mistakes, such as failing to simplify the equation or not combining like terms.