Solve The Equation For { X $} : : : { X = 2(3x - 7) - 6 \}

Mar 13, 2025 by ADMIN 59 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, which is given as: $x = 2(3x - 7) - 6$ . We will break down the solution process into manageable steps, making it easy for readers to understand and follow along.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a linear equation, which means it is in the form of $ax = b$ , where $a$ and $b$ are constants. In this case, the equation is $x = 2(3x - 7) - 6$ . Our goal is to isolate the variable $x$ and find its value.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient $2$ to the terms inside the parentheses. This means we need to multiply $2$ by each term inside the parentheses, which are $3x$ and $-7$ . The result of this operation is:

$x = 2(3x) - 2(7) - 6$

Step 2: Simplify the Equation

Now that we have distributed the coefficient, we can simplify the equation by combining like terms. The terms $2(3x)$ and $-2(7)$ can be simplified as follows:

$x = 6x - 14 - 6$

Step 3: Combine Like Terms

The next step is to combine like terms, which means combining the constant terms $-14$ and $-6$ . The result of this operation is:

$x = 6x - 20$

Step 4: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable $x$ . To do this, we need to get all the terms with $x$ on one side of the equation and the constant terms on the other side. We can do this by subtracting $6x$ from both sides of the equation:

$-5x = -20$

Step 5: Solve for x

The final step is to solve for $x$ . To do this, we need to divide both sides of the equation by $-5$ :

$x = \frac{-20}{-5}$

$x = 4$

Conclusion

In this article, we have solved a linear equation using a step-by-step approach. We started by distributing the coefficient, then simplified the equation, combined like terms, isolated the variable, and finally solved for $x$ . By following these steps, we were able to find the value of $x$ , which is $4$ . This example demonstrates the importance of following a systematic approach when solving linear equations.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS).
Distributing coefficients is a crucial step in solving linear equations.
Combining like terms can help simplify the equation and make it easier to solve.
Isolating the variable is a critical step in solving linear equations.
Solving for $x$ requires careful attention to the signs and operations involved.

Common Mistakes to Avoid

Failing to distribute coefficients can lead to incorrect solutions.
Not combining like terms can result in a more complex equation.
Not isolating the variable can make it difficult to solve for $x$ .
Not paying attention to signs and operations can lead to incorrect solutions.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects.
Engineering: Linear equations are used to design and optimize systems.
Economics: Linear equations are used to model economic systems and make predictions.
Computer Science: Linear equations are used in algorithms and data analysis.

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 enough to just follow a formula. That's why we've put together this Q&A guide to help you tackle common questions and challenges when solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form of ax = b, where a and b are constants.

Q: What is the order of operations (PEMDAS)?

A: PEMDAS is a mnemonic device that helps you remember the order of operations:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I distribute coefficients?

A: To distribute a coefficient, multiply the coefficient by each term inside the parentheses. For example, if you have 2(3x - 7), you would multiply 2 by 3x and -7, resulting in 6x - 14.

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, x = 2(3x - 7) - 6 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve for x when there are fractions involved?

A: To solve for x when there are fractions involved, multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make it easier to solve for x.

Q: What is the importance of isolating the variable?

A: Isolating the variable is crucial when solving linear equations. It allows you to focus on the variable and make it the subject of the equation, making it easier to solve for x.

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 math and be able to solve the equation by hand. This will help you develop problem-solving skills and understand the concept of linear equations.

Q: How do I check my solution?

A: To check your solution, plug the value of x 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 distribute coefficients
Not combining like terms
Not isolating the variable
Not paying attention to signs and operations
Not checking the solution

Conclusion

Solving linear equations can be a challenging task, but with practice and patience, you can master it. By following the steps outlined in this Q&A guide, you'll be able to tackle common questions and challenges when solving linear equations. Remember to always check your solution and be mindful of common mistakes to avoid. Happy solving!