Solve For K K K .${ \frac{8}{k-7} = \frac{2}{5} }$

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Introduction

In algebra, solving for a variable means isolating it on one side of the equation. This is a crucial skill in mathematics, as it allows us to find the value of the variable and make predictions or conclusions based on the equation. In this article, we will focus on solving for $k$ in the equation $\frac{8}{k-7} = \frac{2}{5}$.

Understanding the Equation

Before we dive into solving for $k$, let's take a closer look at the equation. We have a fraction on the left-hand side, with $8$ as the numerator and $k-7$ as the denominator. On the right-hand side, we have a fraction with $2$ as the numerator and $5$ as the denominator. Our goal is to isolate $k$ on one side of the equation.

Step 1: Cross-Multiplying

To solve for $k$, we can start by cross-multiplying the two fractions. This means multiplying the numerator of the left-hand fraction by the denominator of the right-hand fraction, and vice versa. This gives us:

$$8 \cdot 5 = (k-7) \cdot 2$$

Step 2: Simplifying the Equation

Now that we have cross-multiplied, we can simplify the equation by multiplying the numbers together. This gives us:

$$40 = 2k - 14$$

Step 3: Adding 14 to Both Sides

Next, we want to get rid of the negative term on the right-hand side. We can do this by adding $14$ to both sides of the equation. This gives us:

$$40 + 14 = 2k - 14 + 14$$

Step 4: Simplifying the Equation

Now that we have added $14$ to both sides, we can simplify the equation by combining like terms. This gives us:

$$54 = 2k$$

Step 5: Dividing Both Sides by 2

Finally, we want to isolate $k$ on one side of the equation. We can do this by dividing both sides of the equation by $2$. This gives us:

$$\frac{54}{2} = \frac{2k}{2}$$

Step 6: Simplifying the Equation

Now that we have divided both sides by $2$, we can simplify the equation by canceling out the common factor. This gives us:

$$27 = k$$

Conclusion

And there you have it! We have successfully solved for $k$ in the equation $\frac{8}{k-7} = \frac{2}{5}$. By following the steps outlined above, we were able to isolate $k$ on one side of the equation and find its value. This is just one example of how to solve for a variable in an equation, and there are many other techniques and strategies that can be used depending on the specific equation and situation.

Common Mistakes to Avoid

When solving for a variable, it's easy to make mistakes. Here are a few common mistakes to avoid:

• Not following the order of operations: When working with equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the calculations correctly.
• Not simplifying the equation: Simplifying the equation can help you avoid mistakes and make it easier to solve for the variable.
• Not checking your work: It's essential to check your work to ensure that you have solved for the variable correctly.

Real-World Applications

Solving for a variable is a crucial skill in mathematics, and it has many real-world applications. Here are a few examples:

• Science and engineering: In science and engineering, solving for a variable is often used to model real-world phenomena and make predictions.
• Finance: In finance, solving for a variable is used to calculate interest rates, investment returns, and other financial metrics.
• Computer programming: In computer programming, solving for a variable is used to write algorithms and solve problems.

Conclusion

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Introduction

In our previous article, we walked through the steps to solve for $k$ in the equation $\frac{8}{k-7} = \frac{2}{5}$. Now, we're going to take it a step further and answer some of the most frequently asked questions about solving for a variable.

Q: What is the first step in solving for a variable?

A: The first step in solving for a variable is to read the equation carefully and understand what is being asked. In this case, we are solving for $k$, which means we need to isolate $k$ on one side of the equation.

Q: What is cross-multiplying, and why is it important?

A: Cross-multiplying is a technique used to eliminate the fractions in an equation. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. This is an important step in solving for a variable because it allows us to work with whole numbers instead of fractions.

Q: How do I know when to add or subtract a number from both sides of the equation?

A: When working with equations, it's essential to add or subtract the same value from both sides of the equation to maintain the equality. For example, if we have the equation $x + 3 = 5$, we can subtract 3 from both sides to get $x = 2$. Similarly, if we have the equation $x - 2 = 3$, we can add 2 to both sides to get $x = 5$.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation $x + 2 = 5$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula, which is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the order of operations, and why is it important?

A: The order of operations is a set of rules that tells us which operations to perform first when working with mathematical expressions. The order of operations is:

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.

Q: How do I check my work when solving for a variable?

A: When solving for a variable, it's essential to check your work to ensure that you have solved for the variable correctly. Here are a few ways to check your work:

• Plug the value of the variable back into the original equation to see if it is true.
• Use a calculator to check your work.
• Ask a friend or classmate to check your work.

Conclusion

Solving for a variable is a crucial skill in mathematics that has many real-world applications. By following the steps outlined in this article and answering the frequently asked questions, you can become proficient in solving for variables and apply this skill to real-world problems. Remember to always check your work and use the order of operations to ensure that you are performing the calculations correctly.