Solve For $u$.$\frac{12}{u+8}=\frac{6}{5}$Simplify Your Answer As Much As Possible.

Introduction

Rational equations are a type of algebraic equation that involves fractions with variables in the numerator or denominator. Solving for the variable in a rational equation can be a challenging task, but with the right approach, it can be done. In this article, we will solve for the variable u in the rational equation $\frac{12}{u+8}=\frac{6}{5}$.

Understanding the Equation

The given equation is $\frac{12}{u+8}=\frac{6}{5}$. To solve for u, we need to isolate the variable on one side of the equation. The first step is to cross-multiply the fractions, which means multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

Cross-Multiplying

Cross-multiplying the fractions gives us:

$$12 \cdot 5 = 6 \cdot (u+8)$$

Simplifying the equation, we get:

$$60 = 6u + 48$$

Isolating the Variable

To isolate the variable u, we need to get rid of the constant term on the same side of the equation. We can do this by subtracting 48 from both sides of the equation.

$$60 - 48 = 6u + 48 - 48$$

Simplifying the equation, we get:

$$12 = 6u$$

Solving for u

Now that we have isolated the variable u, we can solve for it by dividing both sides of the equation by 6.

$$\frac{12}{6} = \frac{6u}{6}$$

Simplifying the equation, we get:

$$2 = u$$

Conclusion

In this article, we solved for the variable u in the rational equation $\frac{12}{u+8}=\frac{6}{5}$. We started by cross-multiplying the fractions, then isolated the variable u by subtracting 48 from both sides of the equation. Finally, we solved for u by dividing both sides of the equation by 6. The solution to the equation is $u = 2$.

Example Use Cases

Rational equations are used in a variety of real-world applications, including:

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

Tips and Tricks

When solving rational equations, it's essential to follow the order of operations (PEMDAS):

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.

By following these steps and using the correct techniques, you can solve even the most challenging rational equations.

Common Mistakes to Avoid

When solving rational equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not cross-multiplying: Failing to cross-multiply the fractions can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable can make it difficult to solve for the variable.
• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Conclusion

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Introduction

In our previous article, we solved for the variable u in the rational equation $\frac{12}{u+8}=\frac{6}{5}$. In this article, we will answer some common questions that students often have when solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that involves fractions with variables in the numerator or denominator. Rational equations can be used to model real-world situations, such as the motion of objects under the influence of forces or the design of electrical circuits.

Q: How do I know when to cross-multiply?

A: You should cross-multiply when you have two fractions with variables in the numerator or denominator. Cross-multiplying is a way of eliminating the fractions and making it easier to solve for the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

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 isolate the variable?

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

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

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

• Not cross-multiplying: Failing to cross-multiply the fractions can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable can make it difficult to solve for the variable.
• Not following the order of operations (PEMDAS): Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.

Q: Can you give an example of a rational equation?

A: Here is an example of a rational equation:

$$\frac{x+2}{x-3}=\frac{2}{5}$$

To solve for x, you would need to cross-multiply, isolate the variable, and follow the order of operations (PEMDAS).

Q: How do I know if a rational equation has a solution?

A: To determine if a rational equation has a solution, you need to check if the equation is true for any value of the variable. If the equation is true for any value of the variable, then the equation has a solution.

Q: Can you give an example of a rational equation with no solution?

A: Here is an example of a rational equation with no solution:

$$\frac{x+2}{x-3}=\frac{2}{0}$$

This equation has no solution because the denominator is zero, which means that the equation is undefined.

Conclusion

Solving rational equations can be a challenging task, but with the right approach, it can be done. By following the steps outlined in this article and avoiding common mistakes, you can solve for the variable in a rational equation. Remember to cross-multiply, isolate the variable, and follow the order of operations (PEMDAS) to ensure accurate and reliable solutions.