If $-\frac{3}{a-3} = \frac{3}{a+2}$, Then $a =$?A. -3 B. -2 C. $\frac{1}{2}$ D. 2

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Introduction

In this article, we will be solving a mathematical equation involving fractions. The given equation is $-\frac{3}{a-3} = \frac{3}{a+2}$. Our goal is to find the value of $a$ that satisfies this equation. We will use algebraic techniques to solve for $a$.

Step 1: Multiply both sides by the least common multiple (LCM) of the denominators

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. The LCM of $(a-3)$ and $(a+2)$ is $(a-3)(a+2)$. Multiplying both sides by this expression, we get:

$$-\frac{3}{a-3} \cdot (a-3)(a+2) = \frac{3}{a+2} \cdot (a-3)(a+2)$$

Step 3: Simplify the equation

Simplifying the equation, we get:

$$-3(a+2) = 3(a-3)$$

Step 4: Expand and simplify

Expanding and simplifying the equation, we get:

$$-3a - 6 = 3a - 9$$

Step 5: Combine like terms

Combining like terms, we get:

$$-6 = 6a - 9$$

Step 6: Add 9 to both sides

Adding 9 to both sides, we get:

$$3 = 6a$$

Step 7: Divide both sides by 6

Dividing both sides by 6, we get:

$$\frac{1}{2} = a$$

Conclusion

Therefore, the value of $a$ that satisfies the equation $-\frac{3}{a-3} = \frac{3}{a+2}$ is $\frac{1}{2}$.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.

Discussion

This problem involves solving a linear equation with fractions. The key steps involved in solving this equation are multiplying both sides by the LCM of the denominators, simplifying the equation, and combining like terms. The solution to this equation is $\frac{1}{2}$.

Related Problems

If you are looking for more problems like this, you can try solving the following equations:

• $$\frac{1}{x+2} = \frac{2}{x-3}$$

• $$\frac{3}{x-1} = \frac{2}{x+2}$$

• $$\frac{1}{x+1} = \frac{3}{x-2}$$

These problems involve solving linear equations with fractions, and the solution techniques are similar to the one used in this article.

Tips and Tricks

When solving linear equations with fractions, it is often helpful to multiply both sides by the LCM of the denominators. This can help eliminate the fractions and make the equation easier to solve. Additionally, be sure to combine like terms and simplify the equation as much as possible.

Common Mistakes

When solving linear equations with fractions, it is easy to make mistakes. Some common mistakes include:

• Not multiplying both sides by the LCM of the denominators
• Not combining like terms
• Not simplifying the equation enough

To avoid these mistakes, be sure to carefully read the equation and follow the solution steps carefully.

Real-World Applications

Linear equations with fractions can be used to model real-world problems. For example, suppose you are a manager at a company and you need to determine the cost of producing a certain number of units. If the cost of producing one unit is $x$ and the number of units produced is $y$, then the total cost can be modeled by the equation:

$$\frac{1}{x} = \frac{y}{c}$$

where $c$ is the total cost. Solving this equation for $x$, we get:

$$x = \frac{c}{y}$$

This equation can be used to determine the cost of producing a certain number of units.

Conclusion

In conclusion, solving linear equations with fractions involves multiplying both sides by the LCM of the denominators, simplifying the equation, and combining like terms. The solution to the equation $-\frac{3}{a-3} = \frac{3}{a+2}$ is $\frac{1}{2}$. This problem can be used to model real-world problems, such as determining the cost of producing a certain number of units.

Introduction

In our previous article, we solved a linear equation with fractions using algebraic techniques. In this article, we will answer some common questions that students often have when solving linear equations with fractions.

Q: What is the least common multiple (LCM) of two expressions?

A: The least common multiple (LCM) of two expressions is the smallest expression that is a multiple of both expressions. For example, the LCM of $(a-3)$ and $(a+2)$ is $(a-3)(a+2)$.

Q: Why do we need to multiply both sides of the equation by the LCM of the denominators?

A: We need to multiply both sides of the equation by the LCM of the denominators to eliminate the fractions. This is because the LCM is the smallest expression that is a multiple of both denominators.

Q: How do we simplify the equation after multiplying both sides by the LCM?

A: After multiplying both sides by the LCM, we can simplify the equation by combining like terms. This involves adding or subtracting the same term to both sides of the equation.

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

A: Some common mistakes to avoid when solving linear equations with fractions include:

• Not multiplying both sides by the LCM of the denominators
• Not combining like terms
• Not simplifying the equation enough

Q: Can linear equations with fractions be used to model real-world problems?

A: Yes, linear equations with fractions can be used to model real-world problems. For example, suppose you are a manager at a company and you need to determine the cost of producing a certain number of units. If the cost of producing one unit is $x$ and the number of units produced is $y$, then the total cost can be modeled by the equation:

$$\frac{1}{x} = \frac{y}{c}$$

where $c$ is the total cost. Solving this equation for $x$, we get:

$$x = \frac{c}{y}$$

This equation can be used to determine the cost of producing a certain number of units.

Q: How do we determine the LCM of two expressions?

A: To determine the LCM of two expressions, we can list the multiples of each expression and find the smallest multiple that is common to both expressions.

Q: Can we use other methods to solve linear equations with fractions?

A: Yes, we can use other methods to solve linear equations with fractions, such as using the distributive property or using a calculator to simplify the equation.

Q: What are some real-world applications of linear equations with fractions?

A: Some real-world applications of linear equations with fractions include:

• Determining the cost of producing a certain number of units
• Modeling the growth of a population
• Calculating the interest on a loan
• Determining the amount of a discount on a purchase

Conclusion

In conclusion, solving linear equations with fractions involves multiplying both sides by the LCM of the denominators, simplifying the equation, and combining like terms. By following these steps and avoiding common mistakes, we can solve linear equations with fractions and apply them to real-world problems.

Final Tips

• Always multiply both sides of the equation by the LCM of the denominators to eliminate the fractions.
• Simplify the equation as much as possible by combining like terms.
• Use a calculator to check your work and ensure that the equation is true.
• Practice solving linear equations with fractions to become more confident and proficient in your skills.

Related Articles

• Solving Linear Equations with Fractions: A Step-by-Step Guide
• Linear Equations with Fractions: Real-World Applications
• Common Mistakes to Avoid When Solving Linear Equations with Fractions

Resources

• Khan Academy: Solving Linear Equations with Fractions
• Mathway: Solving Linear Equations with Fractions
• Wolfram Alpha: Solving Linear Equations with Fractions