Forgetting to Multiply Both Sides

Oct 13, 2025 by magicalceylontours.com 33 views

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. In this article, we will focus on solving a specific equation: $\frac{(x+8)}{(x-3)} = 5$ . This equation involves fractions and variables, making it a bit more challenging than a simple linear equation. However, with a step-by-step approach, we can break it down and find the solution.

Understanding the Equation

The given equation is $\frac{(x+8)}{(x-3)} = 5$ . To solve this equation, we need to isolate the variable $x$ . The equation involves a fraction, which means we need to get rid of the denominator to solve for $x$ .

Key Concepts

Fractions: A fraction is a way of representing a part of a whole. In this equation, the fraction $\frac{(x+8)}{(x-3)}$ represents the ratio of the numerator $(x+8)$ to the denominator $(x-3)$ .
Equations: An equation is a statement that two expressions are equal. In this case, the equation $\frac{(x+8)}{(x-3)} = 5$ states that the fraction $\frac{(x+8)}{(x-3)}$ is equal to $5$ .

Step 1: Multiply Both Sides by the Denominator

To get rid of the fraction, we can multiply both sides of the equation by the denominator $(x-3)$ . This will eliminate the fraction and allow us to solve for $x$ .

$\frac{(x+8)}{(x-3)} \cdot (x-3) = 5 \cdot (x-3)$

Simplifying the Equation

When we multiply both sides by the denominator, we get:

$x+8 = 5(x-3)$

Step 2: Distribute the 5 on the Right Side

To simplify the equation further, we can distribute the $5$ on the right side.

$x+8 = 5x - 15$

Combining Like Terms

Now, we can combine like terms on the right side.

$x+8 = 5x - 15$

Step 3: Subtract $x$ from Both Sides

To isolate the variable $x$ , we can subtract $x$ from both sides.

$8 = 4x - 15$

Adding 15 to Both Sides

Next, we can add $15$ to both sides to get rid of the negative term.

$23 = 4x$

Step 4: Divide Both Sides by 4

Finally, we can divide both sides by $4$ to solve for $x$ .

$x = \frac{23}{4}$

Simplifying the Solution

The solution to the equation is $x = \frac{23}{4}$ .

Conclusion

Solving the equation $\frac{(x+8)}{(x-3)} = 5$ requires a step-by-step approach. By multiplying both sides by the denominator, distributing the $5$ on the right side, combining like terms, subtracting $x$ from both sides, adding $15$ to both sides, and finally dividing both sides by $4$ , we can find the solution to the equation. The solution is $x = \frac{23}{4}$ .

Real-World Applications

Solving equations like this one has many real-world applications. For example, in physics, equations like this one can be used to model the motion of objects. In finance, equations like this one can be used to calculate interest rates. In engineering, equations like this one can be used to design and optimize systems.

Tips and Tricks

Check Your Work: When solving equations, it's essential to check your work to ensure that the solution is correct.
Use a Calculator: If you're having trouble solving an equation, try using a calculator to check your work.
Practice, Practice, Practice: The more you practice solving equations, the more comfortable you'll become with the process.

Common Mistakes

Forgetting to Multiply Both Sides: When solving equations with fractions, it's essential to multiply both sides by the denominator to eliminate the fraction.
Not Distributing the Coefficient: When solving equations with coefficients, it's essential to distribute the coefficient to all terms on the right side.
Not Checking Your Work: When solving equations, it's essential to check your work to ensure that the solution is correct.

Final Thoughts

Solving equations like $\frac{(x+8)}{(x-3)} = 5$ requires a step-by-step approach. By following these steps and practicing regularly, you can become more comfortable with solving equations and apply this knowledge to real-world problems. Remember to check your work, use a calculator when needed, and practice regularly to become proficient in solving equations.

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Introduction

In our previous article, we solved the equation $\frac{(x+8)}{(x-3)} = 5$ using a step-by-step approach. However, we understand that sometimes, it's not enough to just provide a solution. You may have questions about the process, the steps involved, or the concepts used. That's why we've created this Q&A guide to help you better understand the solution and the concepts involved.

Q: What is the first step in solving the equation $\frac{(x+8)}{(x-3)} = 5$ ?

A: The first step in solving the equation is to multiply both sides by the denominator $(x-3)$ . This will eliminate the fraction and allow us to solve for $x$ .

Q: Why do we need to multiply both sides by the denominator?

A: We need to multiply both sides by the denominator to eliminate the fraction. This is because the fraction is equal to $5$ , and we want to get rid of the fraction to solve for $x$ .

Q: What happens if we forget to multiply both sides by the denominator?

A: If we forget to multiply both sides by the denominator, we will be left with an equation that still contains a fraction. This will make it difficult to solve for $x$ , and we may end up with an incorrect solution.

Q: How do we distribute the $5$ on the right side of the equation?

A: To distribute the $5$ on the right side, we need to multiply the $5$ by each term inside the parentheses. This will give us $5x - 15$ .

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to combine like terms on the right side. This will give us $x + 8 = 5x - 15$ .

Q: Why do we need to combine like terms?

A: We need to combine like terms to simplify the equation and make it easier to solve for $x$ .

Q: What happens if we forget to combine like terms?

A: If we forget to combine like terms, we will be left with an equation that is difficult to solve. This will make it harder to find the correct solution for $x$ .

Q: How do we solve for $x$ ?

A: To solve for $x$ , we need to isolate the variable $x$ on one side of the equation. This can be done by subtracting $x$ from both sides, adding $15$ to both sides, and finally dividing both sides by $4$ .

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = \frac{23}{4}$ .

Q: Why is it essential to check our work when solving equations?

A: It's essential to check our work when solving equations to ensure that the solution is correct. This can be done by plugging the solution back into the original equation and verifying that it's true.

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

A: Some common mistakes to avoid when solving equations include forgetting to multiply both sides by the denominator, not distributing the coefficient, and not checking our work.

Q: How can we practice solving equations to become more proficient?

A: We can practice solving equations by working through examples, using online resources, and seeking help from teachers or tutors.

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

A: Solving equations has many real-world applications, including physics, finance, and engineering. It can be used to model the motion of objects, calculate interest rates, and design and optimize systems.

Q: How can we apply the concepts learned in this article to real-world problems?

A: We can apply the concepts learned in this article to real-world problems by using the steps outlined in the solution to solve equations that involve fractions and variables.

Q: What are some tips and tricks for solving equations?

A: Some tips and tricks for solving equations include checking our work, using a calculator when needed, and practicing regularly to become more proficient.

Conclusion

Solving the equation $\frac{(x+8)}{(x-3)} = 5$ requires a step-by-step approach. By following these steps and practicing regularly, you can become more comfortable with solving equations and apply this knowledge to real-world problems. Remember to check your work, use a calculator when needed, and practice regularly to become proficient in solving equations.