Solve The Equation:$\[ \frac{x+5}{x+2} = \frac{3}{x+2} \\]Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice:A. \[$x =\$\] \[$\square\$\] (Use A Comma To Separate Answers As

Introduction

In mathematics, solving equations is a crucial skill that helps us find the value of unknown variables. In this article, we will focus on solving a specific equation involving fractions. The equation is given as:

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

Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Identify the Common Denominator

The first step in solving this equation is to identify the common denominator, which is the denominator that appears in both fractions. In this case, the common denominator is $x+2$. We can rewrite the equation as:

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

Step 2: Eliminate the Common Denominator

Since the common denominator is the same in both fractions, we can eliminate it by multiplying both sides of the equation by $x+2$. This gives us:

$$(x+5) = 3$$

Step 3: Solve for $x$

Now that we have eliminated the common denominator, we can solve for $x$. To do this, we need to isolate $x$ on one side of the equation. We can do this by subtracting 5 from both sides of the equation:

$$x = 3 - 5$$

Step 4: Simplify the Equation

Now that we have isolated $x$, we can simplify the equation by evaluating the expression on the right-hand side:

$$x = -2$$

Conclusion

In this article, we have solved the equation $\frac{x+5}{x+2} = \frac{3}{x+2}$ by identifying the common denominator, eliminating it, and solving for $x$. The final answer is $x = -2$.

Discussion

This equation is a classic example of a rational equation, which involves fractions with variables in the numerator and denominator. Solving rational equations requires careful attention to the common denominator and the properties of fractions.

Tips and Tricks

When solving rational equations, it's essential to identify the common denominator and eliminate it by multiplying both sides of the equation by that denominator. This will help you avoid unnecessary complications and make the solution process more straightforward.

Common Mistakes

One common mistake when solving rational equations is to forget to eliminate the common denominator. This can lead to incorrect solutions and unnecessary complications. Always make sure to eliminate the common denominator before solving for the variable.

Real-World Applications

Rational equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, rational equations are used to model the motion of objects and the behavior of electrical circuits. In engineering, rational equations are used to design and optimize systems, such as bridges and buildings. In economics, rational equations are used to model the behavior of markets and the impact of policy decisions.

Conclusion

In conclusion, solving rational equations requires careful attention to the common denominator and the properties of fractions. By following the steps outlined in this article, you can solve rational equations and apply the concepts to real-world problems. Remember to identify the common denominator, eliminate it, and solve for the variable to get the correct solution.

Final Answer

$$

Introduction

In our previous article, we solved the equation $\frac{x+5}{x+2} = \frac{3}{x+2}$ by identifying the common denominator, eliminating it, and solving for $x$. In this article, we will answer some frequently asked questions about solving rational equations.

Q: What is a rational equation?

A: A rational equation is an equation that involves fractions with variables in the numerator and denominator. Rational equations can be solved by identifying the common denominator, eliminating it, and solving for the variable.

Q: How do I identify the common denominator?

A: To identify the common denominator, look for the denominator that appears in both fractions. In the equation $\frac{x+5}{x+2} = \frac{3}{x+2}$, the common denominator is $x+2$.

Q: What if the common denominator is not obvious?

A: If the common denominator is not obvious, try factoring the denominators to see if you can find a common factor. For example, in the equation $\frac{x+5}{x+2} = \frac{3}{x+1}$, the common denominator is not obvious. However, if we factor the denominators, we get $(x+2)(x+1)$, which is the common denominator.

Q: How do I eliminate the common denominator?

A: To eliminate the common denominator, multiply both sides of the equation by the common denominator. For example, in the equation $\frac{x+5}{x+2} = \frac{3}{x+2}$, we can eliminate the common denominator by multiplying both sides by $x+2$.

Q: What if the equation has multiple variables?

A: If the equation has multiple variables, you may need to use algebraic techniques such as substitution or elimination to solve for the variables. For example, in the equation $\frac{x+5}{x+2} = \frac{3y}{x+2}$, we can use substitution to solve for $x$ and $y$.

Q: Can I use a calculator to solve rational equations?

A: Yes, you can use a calculator to solve rational equations. However, keep in mind that calculators may not always give you the exact solution, and you may need to use algebraic techniques to verify the solution.

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

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

• Forgetting to eliminate the common denominator
• Not checking for extraneous solutions
• Not using algebraic techniques to solve for multiple variables
• Not verifying the solution using a calculator or algebraic techniques

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

A: Rational equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, rational equations are used to model the motion of objects and the behavior of electrical circuits. In engineering, rational equations are used to design and optimize systems, such as bridges and buildings. In economics, rational equations are used to model the behavior of markets and the impact of policy decisions.

Conclusion

In conclusion, solving rational equations requires careful attention to the common denominator and the properties of fractions. By following the steps outlined in this article, you can solve rational equations and apply the concepts to real-world problems. Remember to identify the common denominator, eliminate it, and solve for the variable to get the correct solution.

Final Answer

The final answer is: $\boxed{-2}$