Enter The Values For The Highlighted Variables To Complete The Steps To Find The Sum:$[ \begin{aligned} \frac{3x}{2x-6} + \frac{9}{6-2x} &= \frac{3x}{2x-6} + \frac{9}{a(2x-6)} \ &= \frac{3x}{2x-6} + \frac{b}{2x-6} \ &= \frac{3x-c}{2x-6} \ &=

Introduction

In this article, we will guide you through the process of solving a complex equation involving fractions. The equation is given as $\frac{3x}{2x-6} + \frac{9}{6-2x} = \frac{3x}{2x-6} + \frac{9}{a(2x-6)}$. Our goal is to find the values of the highlighted variables to complete the steps and find the sum.

Step 1: Simplify the Equation

To simplify the equation, we need to find a common denominator for the two fractions. The common denominator is $2x-6$. We can rewrite the second fraction as $\frac{9}{a(2x-6)} = \frac{9}{2x-6}$. Now, we can add the two fractions together.

$\frac{3x}{2x-6} + \frac{9}{2x-6} = \frac{3x+9}{2x-6}$

Step 2: Factor the Numerator

The numerator of the fraction is $3x+9$. We can factor out the greatest common factor, which is $3$. So, we have:

$\frac{3x+9}{2x-6} = \frac{3(x+3)}{2x-6}$

Step 3: Find the Value of $a$

We are given that $\frac{9}{a(2x-6)} = \frac{9}{2x-6}$. We can equate the denominators to find the value of $a$.

$a(2x-6) = 2x-6$

We can divide both sides by $2x-6$ to get:

$a = 1$

Step 4: Find the Value of $b$

We are given that $\frac{3x}{2x-6} + \frac{b}{2x-6} = \frac{3x-c}{2x-6}$. We can equate the numerators to find the value of $b$.

$3x+b = 3x-c$

We can subtract $3x$ from both sides to get:

$b = -c$

Step 5: Find the Value of $c$

We are given that $\frac{3x}{2x-6} + \frac{b}{2x-6} = \frac{3x-c}{2x-6}$. We can equate the numerators to find the value of $c$.

$3x+b = 3x-c$

We can add $c$ to both sides to get:

$3x+b+c = 3x$

We can subtract $3x$ from both sides to get:

$b+c = 0$

Step 6: Solve for $x$

We are given that $\frac{3x}{2x-6} + \frac{9}{6-2x} = \frac{3x}{2x-6} + \frac{9}{a(2x-6)}$. We can substitute the values of $a$ and $b$ into the equation.

$\frac{3x}{2x-6} + \frac{9}{2x-6} = \frac{3x}{2x-6} + \frac{b}{2x-6}$

We can simplify the equation to get:

$\frac{3x+9}{2x-6} = \frac{3x+b}{2x-6}$

We can equate the numerators to get:

$3x+9 = 3x+b$

We can subtract $3x$ from both sides to get:

$9 = b$

We can substitute the value of $b$ into the equation to get:

$\frac{3x}{2x-6} + \frac{9}{2x-6} = \frac{3x}{2x-6} + \frac{9}{2x-6}$

We can simplify the equation to get:

$\frac{3x+9}{2x-6} = \frac{3x+9}{2x-6}$

We can equate the numerators to get:

$3x+9 = 3x+9$

We can subtract $3x$ from both sides to get:

$9 = 9$

Conclusion

In this article, we have guided you through the process of solving a complex equation involving fractions. We have simplified the equation, factored the numerator, found the values of $a$ and $b$, and solved for $x$. The final answer is $x = \boxed{3}$.

Discussion

This equation is a great example of how to simplify and solve complex equations involving fractions. The key steps are to find a common denominator, factor the numerator, and equate the numerators. With these steps, you can solve even the most complex equations.

Additional Resources

If you are struggling with solving equations involving fractions, here are some additional resources that may help:

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

Final Thoughts

Q: What is the first step in solving an equation with fractions?

A: The first step in solving an equation with fractions is to find a common denominator for the fractions. This will allow you to add or subtract the fractions easily.

Q: How do I find a common denominator?

A: To find a common denominator, you need to identify the denominators of the fractions and find the least common multiple (LCM) of those denominators. For example, if you have two fractions with denominators of 4 and 6, the LCM of 4 and 6 is 12.

Q: What is the next step after finding a common denominator?

A: After finding a common denominator, you can rewrite each fraction with the common denominator. This will allow you to add or subtract the fractions easily.

Q: How do I add or subtract fractions with different denominators?

A: To add or subtract fractions with different denominators, you need to find a common denominator and rewrite each fraction with the common denominator. Then, you can add or subtract the numerators while keeping the common denominator the same.

Q: What is the difference between adding and subtracting fractions?

A: Adding fractions involves combining the numerators while keeping the common denominator the same. Subtracting fractions involves subtracting the numerators while keeping the common denominator the same.

Q: How do I simplify a fraction after adding or subtracting?

A: To simplify a fraction after adding or subtracting, you need to divide the numerator and denominator by their greatest common divisor (GCD). For example, if you have a fraction with a numerator of 12 and a denominator of 18, the GCD of 12 and 18 is 6. So, you can simplify the fraction by dividing both the numerator and denominator by 6.

Q: What is the final step in solving an equation with fractions?

A: The final step in solving an equation with fractions is to isolate the variable. This involves getting the variable by itself on one side of the equation and the constant on the other side.

Q: How do I isolate the variable in an equation with fractions?

A: To isolate the variable in an equation with fractions, you need to get rid of the fractions by multiplying both sides of the equation by the denominator. Then, you can simplify the equation and isolate the variable.

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

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

• Not finding a common denominator
• Not rewriting each fraction with the common denominator
• Not adding or subtracting the numerators correctly
• Not simplifying the fraction after adding or subtracting
• Not isolating the variable correctly

Q: How can I practice solving equations with fractions?

A: You can practice solving equations with fractions by working through examples and exercises in a textbook or online resource. You can also try solving equations with fractions on your own and checking your answers with a calculator or online tool.

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

A: Solving equations with fractions has many real-world applications, including:

• Calculating percentages and proportions
• Solving problems involving rates and ratios
• Working with financial data and investments
• Solving problems involving time and distance

Conclusion

Solving equations with fractions can be challenging, but with practice and patience, you can master this skill. Remember to find a common denominator, rewrite each fraction with the common denominator, add or subtract the numerators, simplify the fraction, and isolate the variable. With these steps, you can solve even the most complex equations with fractions.