Solve The Equation: $\[ \frac{x-16}{x+6} = \frac{3}{5} \\]

Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is $\frac{x-16}{x+6} = \frac{3}{5}$. This equation may seem daunting at first, but with a clear understanding of the steps involved, we can break it down into manageable parts and arrive at a solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what it represents. The equation is a rational equation, which means it contains fractions with variables in the numerator and denominator. The equation is also a linear equation, meaning it can be represented as a straight line on a graph.

Step 1: Cross-Multiplication

To solve the equation, we will start by cross-multiplying. This involves multiplying both sides of the equation by the denominators of the fractions. In this case, we will multiply both sides by $(x+6)$ and $(5)$.

$$\frac{x-16}{x+6} \cdot (x+6) \cdot 5 = \frac{3}{5} \cdot (x+6) \cdot 5$$

This simplifies to:

$$(x-16) \cdot 5 = 3(x+6)$$

Step 2: Distributing the Numbers

Next, we will distribute the numbers on both sides of the equation. This involves multiplying each term inside the parentheses by the number outside the parentheses.

$$5(x-16) = 3(x+6)$$

This simplifies to:

$$5x - 80 = 3x + 18$$

Step 3: Isolating the Variable

Now, we will isolate the variable $x$ by moving all the terms containing $x$ to one side of the equation and the constant terms to the other side.

$$5x - 3x = 18 + 80$$

This simplifies to:

$$2x = 98$$

Step 4: Solving for $x$

Finally, we will solve for $x$ by dividing both sides of the equation by the coefficient of $x$.

$$x = \frac{98}{2}$$

This simplifies to:

$$x = 49$$

Conclusion

In this article, we solved the equation $\frac{x-16}{x+6} = \frac{3}{5}$ step by step. We started by cross-multiplying, then distributed the numbers, isolated the variable, and finally solved for $x$. With a clear understanding of the steps involved, we can confidently say that the solution to the equation is $x = 49$.

Real-World Applications

Solving rational equations like this one has many real-world applications. For example, in physics, we use rational equations to model the motion of objects. In engineering, we use rational equations to design and optimize systems. In finance, we use rational equations to model the behavior of financial markets.

Tips and Tricks

When solving rational equations, it's essential to remember the following tips and tricks:

• Always start by cross-multiplying.
• Distribute the numbers carefully.
• Isolate the variable by moving all the terms containing $x$ to one side of the equation and the constant terms to the other side.
• Solve for $x$ by dividing both sides of the equation by the coefficient of $x$.

By following these tips and tricks, you can confidently solve rational equations like this one and apply them to real-world problems.

Common Mistakes

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

• Not cross-multiplying.
• Not distributing the numbers carefully.
• Not isolating the variable correctly.
• Not solving for $x$ correctly.

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

Conclusion

$$$$$$

Introduction

In our previous article, we solved the equation $\frac{x-16}{x+6} = \frac{3}{5}$ step by step. In this article, we will answer some of the most frequently asked questions about solving rational equations like this one.

Q: What is a rational equation?

A: A rational equation is an equation that contains fractions with variables in the numerator and denominator. Rational equations can be linear or non-linear, and they can be solved using various techniques.

Q: Why do we need to cross-multiply when solving rational equations?

A: Cross-multiplication is a technique used to eliminate the fractions in a rational equation. By multiplying both sides of the equation by the denominators of the fractions, we can simplify the equation and make it easier to solve.

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

A: A linear equation is an equation that can be represented as a straight line on a graph. A non-linear equation, on the other hand, is an equation that cannot be represented as a straight line on a graph. Rational equations can be either linear or non-linear.

Q: How do I know if an equation is linear or non-linear?

A: To determine if an equation is linear or non-linear, you can try graphing it on a coordinate plane. If the graph is a straight line, the equation is linear. If the graph is a curve, the equation is non-linear.

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
• Not distributing the numbers carefully
• Not isolating the variable correctly
• Not solving for $x$ correctly

Q: How do I isolate the variable in a rational equation?

A: To isolate the variable in a rational equation, you need to move all the terms containing the variable to one side of the equation and the constant terms to the other side. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the appropriate values.

Q: What is the final step in solving a rational equation?

A: The final step in solving a rational equation is to solve for $x$ by dividing both sides of the equation by the coefficient of $x$. This will give you the value of $x$ that satisfies the equation.

Q: Can rational equations be used to model real-world problems?

A: Yes, rational equations can be used to model real-world problems. For example, in physics, rational equations are used to model the motion of objects. In engineering, rational equations are used to design and optimize systems. In finance, rational equations are used to model the behavior of financial markets.

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

A: Some real-world applications of rational equations include:

• Modeling the motion of objects in physics
• Designing and optimizing systems in engineering
• Modeling the behavior of financial markets in finance
• Solving problems in computer science and data analysis

Conclusion

In conclusion, solving rational equations requires a clear understanding of the steps involved. By cross-multiplying, distributing the numbers, isolating the variable, and solving for $x$, we can confidently say that the solution to the equation is $x = 49$. With a clear understanding of the steps involved and a few tips and tricks, you can confidently solve rational equations like this one and apply them to real-world problems.