Solve The Equation: 2 ( X + 4 ) 3 = X + 1 − X 3 \frac{2(x+4)}{3} = X + 1 - \frac{x}{3} 3 2 ( X + 4 ) ​ = X + 1 − 3 X ​

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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, 2(x+4)3=x+1−x3\frac{2(x+4)}{3} = x + 1 - \frac{x}{3}, using a step-by-step approach. We will break down the solution into manageable parts, making it easy to understand and follow.

Understanding the Equation

The given equation is 2(x+4)3=x+1−x3\frac{2(x+4)}{3} = x + 1 - \frac{x}{3}. To solve this equation, we need to isolate the variable xx on one side of the equation. The equation involves fractions, so we will need to use algebraic techniques to simplify and solve it.

Step 1: Simplify the Left-Hand Side

The left-hand side of the equation is 2(x+4)3\frac{2(x+4)}{3}. We can simplify this expression by distributing the 22 to the terms inside the parentheses:

2(x+4)3=2x+83\frac{2(x+4)}{3} = \frac{2x + 8}{3}

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation is x+1−x3x + 1 - \frac{x}{3}. We can simplify this expression by combining the like terms:

x+1−x3=3x+3−x3=2x+33x + 1 - \frac{x}{3} = \frac{3x + 3 - x}{3} = \frac{2x + 3}{3}

Combining the Simplified Expressions

Now that we have simplified both sides of the equation, we can combine them:

2x+83=2x+33\frac{2x + 8}{3} = \frac{2x + 3}{3}

Eliminating the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 33:

3×2x+83=3×2x+333 \times \frac{2x + 8}{3} = 3 \times \frac{2x + 3}{3}

This simplifies to:

2x+8=2x+32x + 8 = 2x + 3

Solving for xx

Now that we have eliminated the fractions, we can solve for xx by isolating the variable on one side of the equation. We can do this by subtracting 2x2x from both sides:

2x+8−2x=2x+3−2x2x + 8 - 2x = 2x + 3 - 2x

This simplifies to:

8=38 = 3

However, this is a contradiction, as 88 is not equal to 33. This means that the original equation has no solution.

Conclusion

In this article, we solved the linear equation 2(x+4)3=x+1−x3\frac{2(x+4)}{3} = x + 1 - \frac{x}{3} using a step-by-step approach. We simplified the expressions on both sides of the equation, eliminated the fractions, and solved for xx. However, we found that the equation has no solution, as it leads to a contradiction.

Tips and Tricks

  • When solving linear equations, it's essential to simplify the expressions on both sides of the equation.
  • Eliminating fractions can make the equation easier to solve.
  • Be careful when subtracting or adding terms to both sides of the equation, as this can lead to errors.

Real-World Applications

Linear equations have numerous 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.

Further Reading

For more information on solving linear equations, check out the following resources:

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

Practice Problems

Try solving the following linear equations:

  • x+24=x−1\frac{x + 2}{4} = x - 1
  • 2x+5=3x−22x + 5 = 3x - 2
  • 3x−22=x+1\frac{3x - 2}{2} = x + 1

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach, we can simplify the expressions on both sides of the equation, eliminate fractions, and solve for the variable. However, it's essential to be careful when subtracting or adding terms to both sides of the equation, as this can lead to errors. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

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Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, you should follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I eliminate fractions when solving a linear equation?

A: To eliminate fractions when solving a linear equation, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that is common to two or more numbers. For example, the LCM of 2 and 3 is 6.

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

A: To check your solution to a linear equation, you can plug the solution back into the original equation and see if it is true.

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

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

  • Not following the order of operations
  • Not eliminating fractions
  • Not checking the solution
  • Not being careful when adding, subtracting, multiplying, or dividing both sides of the equation

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be applied to a wide range of real-world problems, including:

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

Q: What are some resources for learning more about solving linear equations?

A: Some resources for learning more about solving linear equations include:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations
  • Online tutorials and videos
  • Textbooks and workbooks

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Working through practice problems
  • Using online resources and tools
  • Creating your own problems and solutions
  • Joining a study group or working with a tutor

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following a step-by-step approach, you can simplify the expressions on both sides of the equation, eliminate fractions, and solve for the variable. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.