Solve For $x$.$\[ \begin{array}{c} 2(x+3) = 4x - 6 \\ x = [?] \end{array} \\]

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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 type of linear equation, which is a simple equation with one variable. We will use the given equation as an example and walk through the steps to solve for the variable x.

The Given Equation

The given equation is:

2(x+3)=4x−62(x+3) = 4x - 6

Our goal is to solve for the variable x.

Step 1: Distribute the Coefficient

The first step in solving this equation is to distribute the coefficient 2 to the terms inside the parentheses.

2(x+3)=2x+62(x+3) = 2x + 6

Now the equation becomes:

2x+6=4x−62x + 6 = 4x - 6

Step 2: Add or Subtract the Same Value to Both Sides

To isolate the variable x, we need to get rid of the constant term on the left side of the equation. We can do this by adding or subtracting the same value to both sides of the equation.

Let's add 6 to both sides of the equation:

2x+6+6=4x−6+62x + 6 + 6 = 4x - 6 + 6

This simplifies to:

2x+12=4x2x + 12 = 4x

Step 3: Subtract the Same Value from Both Sides

Now we need to get rid of the term with the variable x on the right side of the equation. We can do this by subtracting the same value from both sides of the equation.

Let's subtract 2x from both sides of the equation:

2x+12−2x=4x−2x2x + 12 - 2x = 4x - 2x

This simplifies to:

12=2x12 = 2x

Step 4: Divide Both Sides by the Coefficient

Finally, we need to isolate the variable x by dividing both sides of the equation by the coefficient of x.

Let's divide both sides of the equation by 2:

122=2x2\frac{12}{2} = \frac{2x}{2}

This simplifies to:

6=x6 = x

Conclusion

In this article, we solved a simple linear equation using the given equation as an example. We walked through the steps to solve for the variable x, including distributing the coefficient, adding or subtracting the same value to both sides, subtracting the same value from both sides, and dividing both sides by the coefficient.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

  • Always start by distributing the coefficient to the terms inside the parentheses.
  • Use addition or subtraction to get rid of the constant term on the left side of the equation.
  • Use subtraction to get rid of the term with the variable x on the right side of the equation.
  • Finally, divide both sides of the equation by the coefficient of x.

Practice Problems

Here are some practice problems to help you practice solving linear equations:

  1. Solve for x: 3(x-2) = 2x + 5
  2. Solve for x: 2x + 4 = 5x - 3
  3. Solve for x: x + 2 = 3x - 1

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

Introduction

In our previous article, we walked through the steps to solve a simple linear equation. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts or questions you may have. In this article, we'll provide a Q&A guide to help you better understand how to solve 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. In other words, it's an equation that can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

  • The highest power of the variable (usually x) is 1.
  • The equation can be written in the form ax + b = c, where a, b, and c are constants.
  • The equation does not contain any exponents or roots.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to distribute the coefficient to the terms inside the parentheses. This will help you simplify the equation and make it easier to solve.

Q: How do I add or subtract the same value to both sides of an equation?

A: To add or subtract the same value to both sides of an equation, simply add or subtract the value to both sides of the equation. For example, if you have the equation 2x + 3 = 5, you can add 3 to both sides to get 2x + 6 = 8.

Q: How do I subtract the same value from both sides of an equation?

A: To subtract the same value from both sides of an equation, simply subtract the value from both sides of the equation. For example, if you have the equation 2x + 3 = 5, you can subtract 3 from both sides to get 2x = 2.

Q: How do I divide both sides of an equation by a coefficient?

A: To divide both sides of an equation by a coefficient, simply divide both sides of the equation by the coefficient. For example, if you have the equation 2x = 4, you can divide both sides by 2 to get x = 2.

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

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

  • Not distributing the coefficient to the terms inside the parentheses.
  • Not adding or subtracting the same value to both sides of the equation.
  • Not subtracting the same value from both sides of the equation.
  • Not dividing both sides of the equation by the coefficient.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Using online resources, such as Khan Academy or Mathway.
  • Working with a tutor or teacher.
  • Practicing with worksheets or online exercises.
  • Solving real-world problems that involve linear equations.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and practicing with real-world problems, you can become proficient in solving linear equations and apply them to a wide range of fields. Remember to always distribute the coefficient, add or subtract the same value to both sides, subtract the same value from both sides, and divide both sides by the coefficient. With practice and patience, you can become a master of solving linear equations.

Additional Resources

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

Practice Problems

Here are some practice problems to help you practice solving linear equations:

  1. Solve for x: 3(x-2) = 2x + 5
  2. Solve for x: 2x + 4 = 5x - 3
  3. Solve for x: x + 2 = 3x - 1

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and practicing with real-world problems, you can become proficient in solving linear equations and apply them to a wide range of fields. Remember to always distribute the coefficient, add or subtract the same value to both sides, subtract the same value from both sides, and divide both sides by the coefficient. With practice and patience, you can become a master of solving linear equations.