A Student Solves The Following Equation For All Possible Values Of $x$:$\frac{8}{x+2}=\frac{2}{x-4}$His Solution Is As Follows:- Step 1: $8(x-4)=2(x+2$\]- Step 2: $4(x-4)=(x+2$\]- Step 3: $4x-16=x+2$- Step

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Introduction

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we will guide you through the process of solving a specific equation for all possible values of x. We will break down the solution into manageable steps and provide a clear explanation of each step.

The Equation

The given equation is:

8x+2=2x−4\frac{8}{x+2}=\frac{2}{x-4}

This equation involves fractions, and our goal is to solve for x.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is (x+2)(x-4).

8(x-4) = 2(x+2)

This step is crucial in simplifying the equation and making it easier to solve.

Step 2: Distribute and Simplify

Now, we need to distribute the numbers outside the parentheses to the terms inside.

8x - 32 = 2x + 4

Next, we can simplify the equation by combining like terms.

6x - 36 = 0

Step 3: Add 36 to Both Sides

To isolate the term with x, we need to add 36 to both sides of the equation.

6x = 36

Step 4: Divide Both Sides by 6

Finally, we can solve for x by dividing both sides of the equation by 6.

x = 6

Conclusion

In this article, we have walked you through the process of solving a specific equation for all possible values of x. We have broken down the solution into manageable steps and provided a clear explanation of each step. By following these steps, you should be able to solve similar equations with ease.

Common Mistakes to Avoid

When solving equations, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are a few mistakes to watch out for:

  • Not checking the domain: Before solving an equation, make sure to check the domain of the variable. In this case, x cannot be equal to -2 or 4.
  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions. Make sure to combine like terms and eliminate fractions.
  • Not checking the solution: After solving the equation, make sure to check the solution by plugging it back into the original equation.

Real-World Applications

Solving equations has numerous real-world applications in various fields, including:

  • Physics: Equations are used to describe the motion of objects and predict their behavior.
  • Engineering: Equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Equations are used to model economic systems and predict the behavior of markets.

Final Thoughts

Solving equations is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you should be able to solve similar equations with ease. Remember to check the domain, simplify the equation, and check the solution to avoid common mistakes.

Additional Resources

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

  • Mathway: A online math problem solver that can help you solve equations and other math problems.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on solving equations and other math topics.
  • MIT OpenCourseWare: A free online resource that provides lecture notes and practice exercises on solving equations and other math topics.

Q: What is an equation?

A: An equation is a statement that two mathematical expressions are equal. It consists of variables, constants, and mathematical operations.

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

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Add or subtract the same value to both sides of the equation to isolate the variable.
  3. Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Factor the equation, if possible.
  2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
  3. Simplify the equation and solve for the variable.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

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

Q: How do I check my solution?

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

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

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

  • Not checking the domain of the variable.
  • Not simplifying the equation.
  • Not checking the solution.
  • Not following the order of operations.

Q: How do I use equations in real-world applications?

A: Equations are used in a wide range of real-world applications, including:

  • Physics: Equations are used to describe the motion of objects and predict their behavior.
  • Engineering: Equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Equations are used to model economic systems and predict the behavior of markets.

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

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

  • Mathway: A online math problem solver that can help you solve equations and other math problems.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on solving equations and other math topics.
  • MIT OpenCourseWare: A free online resource that provides lecture notes and practice exercises on solving equations and other math topics.

By following these resources and practicing regularly, you should be able to become proficient in solving equations and apply them to real-world problems.