Simplify The Equation Fully. 5 X − 8 = 8 X − 2 \frac{5}{x-8} = \frac{8}{x-2} X − 8 5 ​ = X − 2 8 ​

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Introduction

Simplifying equations is a crucial skill in mathematics, and it's essential to understand how to do it correctly. In this article, we will focus on simplifying a specific type of equation, which is a rational equation. Rational equations involve fractions with variables in the numerator and denominator. We will use the equation 5x−8=8x−2\frac{5}{x-8} = \frac{8}{x-2} as an example to demonstrate the steps involved in simplifying it.

What is a Rational Equation?

A rational equation is an equation that contains one or more fractions with variables in the numerator and denominator. Rational equations can be linear or quadratic, and they can involve one or more variables. In this article, we will focus on simplifying a linear rational equation.

The Equation to Simplify

The equation we will simplify is 5x−8=8x−2\frac{5}{x-8} = \frac{8}{x-2}. This equation involves two fractions with variables in the numerator and denominator. Our goal is to simplify this equation and find the value of x.

Step 1: Cross-Multiply

The first step in simplifying a rational equation is to cross-multiply. Cross-multiplying involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This will eliminate the fractions and give us a new equation.

5x−8=8x−2\frac{5}{x-8} = \frac{8}{x-2}

Cross-multiplying:

5(x−2)=8(x−8)5(x-2) = 8(x-8)

Step 2: Distribute

The next step is to distribute the numbers outside the parentheses to the terms inside. This will give us a new equation with no parentheses.

5(x−2)=8(x−8)5(x-2) = 8(x-8)

Distributing:

5x−10=8x−645x - 10 = 8x - 64

Step 3: Isolate the Variable

The final step is to isolate the variable x. We can do this by moving all the terms involving x to one side of the equation and the constant terms to the other side.

5x−10=8x−645x - 10 = 8x - 64

Subtracting 5x from both sides:

−10=3x−64-10 = 3x - 64

Adding 64 to both sides:

54=3x54 = 3x

Dividing both sides by 3:

18=x18 = x

Conclusion

Simplifying equations is an essential skill in mathematics, and it's crucial to understand how to do it correctly. In this article, we used the equation 5x−8=8x−2\frac{5}{x-8} = \frac{8}{x-2} as an example to demonstrate the steps involved in simplifying it. We cross-multiplied, distributed, and isolated the variable x to find the value of x. By following these steps, you can simplify any rational equation and find the value of the variable.

Tips and Tricks

  • Always cross-multiply when simplifying a rational equation.
  • Distribute the numbers outside the parentheses to the terms inside.
  • Isolate the variable x by moving all the terms involving x to one side of the equation and the constant terms to the other side.
  • Check your work by plugging the value of x back into the original equation.

Common Mistakes

  • Failing to cross-multiply when simplifying a rational equation.
  • Not distributing the numbers outside the parentheses to the terms inside.
  • Not isolating the variable x by moving all the terms involving x to one side of the equation and the constant terms to the other side.

Real-World Applications

Simplifying equations has many real-world applications. For example, in physics, you may need to simplify equations to describe the motion of objects. In engineering, you may need to simplify equations to design and build structures. In economics, you may need to simplify equations to model the behavior of markets.

Conclusion

Introduction

Simplifying equations is a crucial skill in mathematics, and it's essential to understand how to do it correctly. In our previous article, we provided a step-by-step guide on how to simplify a rational equation. In this article, we will answer some frequently asked questions about simplifying equations.

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more fractions with variables in the numerator and denominator. Rational equations can be linear or quadratic, and they can involve one or more variables.

Q: Why do I need to simplify equations?

A: Simplifying equations is essential in mathematics because it helps you to:

  • Solve equations more easily
  • Understand the relationships between variables
  • Make predictions and models in real-world applications

Q: What are the steps involved in simplifying a rational equation?

A: The steps involved in simplifying a rational equation are:

  1. Cross-multiply
  2. Distribute
  3. Isolate the variable x

Q: What is cross-multiplication?

A: Cross-multiplication is the process of multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This will eliminate the fractions and give us a new equation.

Q: What is distribution?

A: Distribution is the process of multiplying the numbers outside the parentheses to the terms inside. This will give us a new equation with no parentheses.

Q: How do I isolate the variable x?

A: To isolate the variable x, you need to move all the terms involving x to one side of the equation and the constant terms to the other side.

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

A: Some common mistakes to avoid when simplifying equations are:

  • Failing to cross-multiply when simplifying a rational equation
  • Not distributing the numbers outside the parentheses to the terms inside
  • Not isolating the variable x by moving all the terms involving x to one side of the equation and the constant terms to the other side

Q: How do I check my work when simplifying equations?

A: To check your work when simplifying equations, you need to plug the value of x back into the original equation and see if it's true.

Q: What are some real-world applications of simplifying equations?

A: Simplifying equations has many real-world applications, such as:

  • Physics: Simplifying equations to describe the motion of objects
  • Engineering: Simplifying equations to design and build structures
  • Economics: Simplifying equations to model the behavior of markets

Q: Can I simplify equations with more than one variable?

A: Yes, you can simplify equations with more than one variable. However, you need to follow the same steps as before, and you may need to use additional techniques, such as substitution or elimination.

Conclusion

Simplifying equations is a crucial skill in mathematics, and it's essential to understand how to do it correctly. By following the steps outlined in this article, you can simplify any rational equation and find the value of the variable. Remember to always cross-multiply, distribute, and isolate the variable x to ensure that you get the correct answer.

Additional Resources

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

Practice Problems

  • Simplify the equation 3x+2=4x−1\frac{3}{x+2} = \frac{4}{x-1}
  • Simplify the equation 2x−3=5x+2\frac{2}{x-3} = \frac{5}{x+2}
  • Simplify the equation 1x+1=2x−2\frac{1}{x+1} = \frac{2}{x-2}