Solve For { X $} . . . { \frac{8x}{9} - \frac{2x}{3} = 4 \}

by ADMIN 60 views

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill to master. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable. We will use the given equation as an example to demonstrate the step-by-step process of solving linear equations.

The Given Equation

The given equation is:

8x9−2x3=4\frac{8x}{9} - \frac{2x}{3} = 4

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation states that the difference between two fractions, 8x9\frac{8x}{9} and 2x3\frac{2x}{3}, is equal to 4. Our goal is to isolate the variable xx and find its value.

Step 1: Simplify the Equation

To simplify the equation, we need to find a common denominator for the two fractions. The least common multiple (LCM) of 9 and 3 is 9. Therefore, we can rewrite the equation as:

8x9−6x9=4\frac{8x}{9} - \frac{6x}{9} = 4

Step 2: Combine Like Terms

Now that we have a common denominator, we can combine the two fractions:

8x−6x9=4\frac{8x - 6x}{9} = 4

Step 3: Simplify the Fraction

We can simplify the fraction by combining the numerators:

2x9=4\frac{2x}{9} = 4

Step 4: Multiply Both Sides by the Reciprocal

To isolate the variable xx, we need to multiply both sides of the equation by the reciprocal of the fraction, which is 92\frac{9}{2}:

x=4×92x = 4 \times \frac{9}{2}

Step 5: Simplify the Expression

We can simplify the expression by multiplying 4 and 92\frac{9}{2}:

x=18x = 18

Conclusion

In this article, we solved a linear equation by isolating the variable xx. We started by simplifying the equation, combining like terms, and multiplying both sides by the reciprocal. Finally, we simplified the expression to find the value of xx. This step-by-step process can be applied to solve any linear equation.

Tips and Tricks

  • When solving linear equations, it's essential to simplify the equation by finding a common denominator.
  • Combining like terms can help simplify the equation and make it easier to solve.
  • Multiplying both sides of the equation by the reciprocal can help isolate the variable.
  • Always check your work by plugging the solution back into the original equation.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, such as the trajectory of a projectile.
  • 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, such as supply and demand curves.

Common Mistakes

  • Failing to simplify the equation by finding a common denominator.
  • Not combining like terms.
  • Not multiplying both sides of the equation by the reciprocal.
  • Not checking the solution by plugging it back into the original equation.

Conclusion

Solving linear equations is a crucial skill in mathematics, and it has numerous real-world applications. By following the step-by-step process outlined in this article, you can master the art of solving linear equations and apply it to various fields. Remember to simplify the equation, combine like terms, multiply both sides by the reciprocal, and check your work to ensure accuracy.

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, and that's where this Q&A guide comes in. We'll address some of the most common questions and concerns that students and professionals may have when it comes to 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 ax + b = c, where a, b, and c are constants.

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, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to find a common denominator for the fractions, if any. Then, combine like terms and simplify the expression.

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

A: The order of operations is:

  1. Simplify the equation by finding a common denominator and combining like terms.
  2. Multiply both sides of the equation by the reciprocal, if necessary.
  3. Add or subtract the same value to both sides of the equation, if necessary.
  4. Divide both sides of the equation by the coefficient of the variable, if necessary.

Q: How do I check my work when solving a linear equation?

A: To check your work, plug the solution back into the original equation and verify that it's true. If the solution satisfies the equation, then you've solved it correctly.

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

A: Some common mistakes to avoid include:

  • Failing to simplify the equation by finding a common denominator.
  • Not combining like terms.
  • Not multiplying both sides of the equation by the reciprocal.
  • Not checking the solution by plugging it back into the original equation.

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 concept and the steps involved in solving linear equations, even if you're using a calculator.

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

A: Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, such as the trajectory of a projectile.
  • 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, such as supply and demand curves.

Q: What are some advanced topics in linear equations?

A: Some advanced topics in linear equations include:

  • Systems of linear equations
  • Matrices and determinants
  • Linear transformations
  • Eigenvalues and eigenvectors

Conclusion

Solving linear equations is a crucial skill in mathematics, and it has numerous real-world applications. By understanding the concept and the steps involved in solving linear equations, you can apply it to various fields and solve complex problems. Remember to simplify the equation, combine like terms, multiply both sides by the reciprocal, and check your work to ensure accuracy.

Additional Resources

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

Final Tips

  • Practice, practice, practice! The more you practice solving linear equations, the more comfortable you'll become with the concept.
  • Use online resources, such as Khan Academy and Mathway, to supplement your learning.
  • Don't be afraid to ask for help if you're struggling with a particular concept or problem.