Solve For \[$ X \$\]:$\[ \frac{x+5}{2x-6} = \frac{5}{2} \\]

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Introduction

Solving equations involving fractions can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving the equation $\frac{x+5}{2x-6} = \frac{5}{2}$ for the variable $x$. This equation involves fractions, and we will use various techniques to simplify and solve it.

Understanding the Equation

The given equation is $\frac{x+5}{2x-6} = \frac{5}{2}$. To solve this equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of $2x-6$ and $2$ is $2(2x-6)$.

Multiplying Both Sides by the LCM

To eliminate the fractions, we multiply both sides of the equation by $2(2x-6)$. This gives us:

$$2(2x-6)\cdot\frac{x+5}{2x-6} = 2(2x-6)\cdot\frac{5}{2}$$

Simplifying the left-hand side, we get:

$$2(x+5) = 2(2x-6)\cdot\frac{5}{2}$$

Simplifying the Equation

Now, we can simplify the right-hand side of the equation:

$$2(x+5) = 5(2x-6)$$

Expanding the right-hand side, we get:

$$2x+10 = 10x-30$$

Isolating the Variable

To isolate the variable $x$, we need to get all the terms involving $x$ on one side of the equation. We can do this by subtracting $2x$ from both sides:

$$10 = 8x-30$$

Adding 30 to Both Sides

Next, we add 30 to both sides of the equation to get:

$$40 = 8x$$

Dividing Both Sides by 8

Finally, we divide both sides of the equation by 8 to solve for $x$:

$$x = \frac{40}{8}$$

Simplifying the Fraction

The fraction $\frac{40}{8}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 8. This gives us:

$$x = \frac{5}{1}$$

Conclusion

In this article, we solved the equation $\frac{x+5}{2x-6} = \frac{5}{2}$ for the variable $x$. We used various techniques, including multiplying both sides by the LCM, simplifying the equation, isolating the variable, and dividing both sides by a coefficient. The final solution is $x = 5$.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. Multiply both sides of the equation by the LCM of the denominators: $2(2x-6)$.
2. Simplify the left-hand side of the equation: $2(x+5)$.
3. Simplify the right-hand side of the equation: $5(2x-6)$.
4. Expand the right-hand side of the equation: $10x-30$.
5. Subtract $2x$ from both sides of the equation: $10 = 8x-30$.
6. Add 30 to both sides of the equation: $40 = 8x$.
7. Divide both sides of the equation by 8: $x = \frac{40}{8}$.
8. Simplify the fraction: $x = \frac{5}{1}$.

Frequently Asked Questions

• What is the least common multiple (LCM) of the denominators? The LCM of $2x-6$ and $2$ is $2(2x-6)$.
• How do I simplify the equation? To simplify the equation, we can multiply both sides by the LCM, simplify the left-hand side, and simplify the right-hand side.
• How do I isolate the variable? To isolate the variable, we can subtract $2x$ from both sides of the equation.
• How do I divide both sides of the equation by a coefficient? To divide both sides of the equation by a coefficient, we can multiply both sides by the reciprocal of the coefficient.

Final Answer

The final answer is: $\boxed{5}$

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Introduction

In our previous article, we solved the equation $\frac{x+5}{2x-6} = \frac{5}{2}$ for the variable $x$. We used various techniques, including multiplying both sides by the LCM, simplifying the equation, isolating the variable, and dividing both sides by a coefficient. In this article, we will answer some frequently asked questions related to the solution of the equation.

Q&A

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

A: The LCM of $2x-6$ and $2$ is $2(2x-6)$.

Q: How do I simplify the equation?

A: To simplify the equation, you can multiply both sides by the LCM, simplify the left-hand side, and simplify the right-hand side.

Q: How do I isolate the variable?

A: To isolate the variable, you can subtract $2x$ from both sides of the equation.

Q: How do I divide both sides of the equation by a coefficient?

A: To divide both sides of the equation by a coefficient, you can multiply both sides by the reciprocal of the coefficient.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 5$.

Q: Can I use other methods to solve the equation?

A: Yes, you can use other methods to solve the equation, such as using algebraic manipulations or using a calculator.

Q: How do I check my answer?

A: To check your answer, you can plug the value of $x$ back into the original equation and see if it is true.

Q: What if I get a different answer?

A: If you get a different answer, it may be due to a mistake in your calculations. Double-check your work and make sure you are following the correct steps.

Tips and Tricks

• Make sure to simplify the equation before isolating the variable.
• Use the correct method to divide both sides of the equation by a coefficient.
• Check your answer by plugging the value of $x$ back into the original equation.
• If you get a different answer, double-check your work and make sure you are following the correct steps.

Common Mistakes

• Failing to simplify the equation before isolating the variable.
• Using the wrong method to divide both sides of the equation by a coefficient.
• Not checking the answer by plugging the value of $x$ back into the original equation.
• Not double-checking the work if a different answer is obtained.

Conclusion

In this article, we answered some frequently asked questions related to the solution of the equation $\frac{x+5}{2x-6} = \frac{5}{2}$. We also provided some tips and tricks for solving equations and common mistakes to avoid. By following the correct steps and checking your answer, you can ensure that you are getting the correct solution to the equation.

Final Answer

The final answer is: $\boxed{5}$