Solve The Equation:${ \frac{10}{a+7}=\frac{6}{7} }$

Introduction

In this article, we will delve into the world of algebra and solve a seemingly complex equation. The equation in question is $\frac{10}{a+7}=\frac{6}{7}$. Our goal is to isolate the variable $a$ and find its value. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Cross-Multiplication

The first step in solving this equation is to eliminate the fractions. We can do this by cross-multiplying, which involves multiplying both sides of the equation by the denominators. In this case, we will multiply both sides by $(a+7)$ and $7$.

$$\frac{10}{a+7}=\frac{6}{7}$$

$$10 \cdot 7 = 6 \cdot (a+7)$$

$$70 = 6a + 42$$

Step 2: Simplifying the Equation

Now that we have eliminated the fractions, we can simplify the equation by combining like terms. We will subtract $42$ from both sides of the equation to isolate the term with the variable.

$$70 - 42 = 6a + 42 - 42$$

$$28 = 6a$$

Step 3: Solving for $a$

The final step is to solve for $a$. We can do this by dividing both sides of the equation by $6$.

$$\frac{28}{6} = \frac{6a}{6}$$

$$\frac{14}{3} = a$$

Conclusion

And there you have it! We have successfully solved the equation $\frac{10}{a+7}=\frac{6}{7}$. By following the steps outlined above, we were able to isolate the variable $a$ and find its value. The final answer is $\boxed{\frac{14}{3}}$.

Real-World Applications

Solving equations like this one has numerous real-world applications. For example, in physics, we often encounter equations that describe the motion of objects. By solving these equations, we can determine the position, velocity, and acceleration of the object. In finance, we use equations to model the growth of investments and make informed decisions about our financial portfolios.

Tips and Tricks

When solving equations like this one, it's essential to follow the order of operations (PEMDAS). This means that we should perform the operations in the following order:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps, we can ensure that we are solving the equation correctly and avoiding any potential errors.

Common Mistakes

When solving equations like this one, there are several common mistakes that we can make. These include:

• Not following the order of operations (PEMDAS)
• Not simplifying the equation correctly
• Not isolating the variable correctly
• Not checking our work for errors

By being aware of these common mistakes, we can take steps to avoid them and ensure that we are solving the equation correctly.

Conclusion

$$$$$$

Introduction

In our previous article, we solved the equation $\frac{10}{a+7}=\frac{6}{7}$ and found the value of $a$ to be $\boxed{\frac{14}{3}}$. However, we know that there are many more questions and concerns that our readers may have. In this article, we will address some of the most frequently asked questions about solving equations like this one.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next.
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: Simplifying an equation involves combining like terms and eliminating any unnecessary operations. To simplify an equation, follow these steps:

1. Combine like terms: Combine any terms that have the same variable and coefficient.
2. Eliminate unnecessary operations: Eliminate any operations that are not necessary to solve the equation.
3. Check your work: Check your work to make sure that you have simplified the equation correctly.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change. In the equation $\frac{10}{a+7}=\frac{6}{7}$, $a$ is a variable and $7$ is a constant.

Q: How do I isolate a variable?

A: Isolating a variable involves getting the variable by itself on one side of the equation. To isolate a variable, follow these steps:

1. Add or subtract the same value to both sides of the equation.
2. Multiply or divide both sides of the equation by the same value.
3. Check your work: Check your work to make sure that you have isolated the variable correctly.

Q: What is the final answer to the equation $\frac{10}{a+7}=\frac{6}{7}$?

A: The final answer to the equation $\frac{10}{a+7}=\frac{6}{7}$ is $\boxed{\frac{14}{3}}$.

Q: Can I use a calculator to solve equations like this one?

A: Yes, you can use a calculator to solve equations like this one. However, it's always a good idea to check your work by hand to make sure that you have solved the equation correctly.

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

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

• Not following the order of operations (PEMDAS)
• Not simplifying the equation correctly
• Not isolating the variable correctly
• Not checking your work for errors

Conclusion

In conclusion, solving equations like $\frac{10}{a+7}=\frac{6}{7}$ requires careful attention to detail and a step-by-step approach. By following the steps outlined above and avoiding common mistakes, you can solve equations like this one with confidence. We hope that this article has provided you with a clear understanding of how to solve equations like this one and has given you the confidence to tackle more complex equations in the future.