Solve The Equation For { A$} . . . {K = 4a + 9ab\} A. { A = K(4 + 9b)$}$B. { A = 4(K + 9b)$}$C. { A = \frac{4 + 9b}{K}$}$D. { A = \frac{K}{4 + 9b}$}$

Introduction

In algebra, solving equations is a crucial skill that helps us find the value of unknown variables. In this article, we will focus on solving the equation $K = 4a + 9ab$ for the variable $a$. We will explore the different options provided and determine the correct solution.

Understanding the Equation

The given equation is $K = 4a + 9ab$. To solve for $a$, we need to isolate the variable $a$ on one side of the equation. The equation involves two variables, $a$ and $b$, and a constant $K$.

Option A: a = K(4 + 9b)

Let's start by analyzing option A: $a = K(4 + 9b)$. To determine if this is the correct solution, we need to substitute the expression for $a$ back into the original equation and check if it holds true.

Substituting $a = K(4 + 9b)$ into the original equation, we get:

$K = 4(K(4 + 9b)) + 9(K(4 + 9b))b$

Expanding and simplifying the equation, we get:

$K = 4K(4 + 9b) + 9K(4 + 9b)b$

$K = 16K + 36Kb + 36Kb + 81Kb^2$

$K = 16K + 72Kb + 81Kb^2$

Subtracting $16K$ from both sides, we get:

$-16K = 72Kb + 81Kb^2$

Dividing both sides by $-16K$, we get:

$1 = -\frac{9}{2}b - \frac{81}{16}b^2$

This equation does not hold true for all values of $b$, so option A is not the correct solution.

Option B: a = 4(K + 9b)

Next, let's analyze option B: $a = 4(K + 9b)$. To determine if this is the correct solution, we need to substitute the expression for $a$ back into the original equation and check if it holds true.

Substituting $a = 4(K + 9b)$ into the original equation, we get:

$K = 4(4(K + 9b)) + 9(4(K + 9b))b$

Expanding and simplifying the equation, we get:

$K = 16(K + 9b) + 36(K + 9b)b$

$K = 16K + 144Kb + 36Kb + 324Kb^2$

$K = 16K + 180Kb + 324Kb^2$

Subtracting $16K$ from both sides, we get:

$-16K = 180Kb + 324Kb^2$

Dividing both sides by $-16K$, we get:

$1 = -\frac{45}{4}b - \frac{81}{4}b^2$

This equation does not hold true for all values of $b$, so option B is not the correct solution.

Option C: a = \frac{4 + 9b}{K}

Next, let's analyze option C: $a = \frac{4 + 9b}{K}$. To determine if this is the correct solution, we need to substitute the expression for $a$ back into the original equation and check if it holds true.

Substituting $a = \frac{4 + 9b}{K}$ into the original equation, we get:

$K = 4\left(\frac{4 + 9b}{K}\right) + 9\left(\frac{4 + 9b}{K}\right)b$

Expanding and simplifying the equation, we get:

$K = \frac{16 + 36b}{K} + \frac{36 + 81b}{K}b$

Multiplying both sides by $K$, we get:

$K^2 = 16 + 36b + 36b + 81b^2$

$K^2 = 16 + 72b + 81b^2$

Subtracting $16$ from both sides, we get:

$K^2 - 16 = 72b + 81b^2$

This equation does not hold true for all values of $b$, so option C is not the correct solution.

Option D: a = \frac{K}{4 + 9b}

Finally, let's analyze option D: $a = \frac{K}{4 + 9b}$. To determine if this is the correct solution, we need to substitute the expression for $a$ back into the original equation and check if it holds true.

Substituting $a = \frac{K}{4 + 9b}$ into the original equation, we get:

$K = 4\left(\frac{K}{4 + 9b}\right) + 9\left(\frac{K}{4 + 9b}\right)b$

Expanding and simplifying the equation, we get:

$K = \frac{4K}{4 + 9b} + \frac{36K}{4 + 9b}b$

Multiplying both sides by $(4 + 9b)$, we get:

$K(4 + 9b) = 4K + 36Kb$

Expanding and simplifying the equation, we get:

$4K + 9Kb = 4K + 36Kb$

Subtracting $4K$ from both sides, we get:

$9Kb = 36Kb$

Dividing both sides by $9Kb$, we get:

$1 = 4$

This equation does not hold true for all values of $b$, so option D is not the correct solution.

Conclusion

After analyzing all the options, we can conclude that none of the options provided is the correct solution to the equation $K = 4a + 9ab$ for the variable $a$. However, we can try to find the correct solution by isolating the variable $a$ on one side of the equation.

To solve for $a$, we can start by subtracting $9ab$ from both sides of the equation:

$K - 9ab = 4a$

Next, we can divide both sides of the equation by $4$:

$\frac{K - 9ab}{4} = a$

Therefore, the correct solution to the equation $K = 4a + 9ab$ for the variable $a$ is:

$a = \frac{K - 9ab}{4}$

This solution can be further simplified by factoring out the common term $b$:

$a = \frac{K}{4 + 9b}$

$$$$

Introduction

In our previous article, we explored the equation $K = 4a + 9ab$ and found the correct solution for the variable $a$. In this article, we will provide a Q&A guide to help you understand the solution and apply it to different scenarios.

Q: What is the correct solution to the equation $K = 4a + 9ab$ for the variable $a$?

A: The correct solution to the equation $K = 4a + 9ab$ for the variable $a$ is:

$a = \frac{K}{4 + 9b}$

Q: How do I apply the solution to a specific problem?

A: To apply the solution to a specific problem, you need to substitute the values of $K$ and $b$ into the equation. For example, if $K = 10$ and $b = 2$, you can substitute these values into the equation:

$a = \frac{10}{4 + 9(2)}$

$a = \frac{10}{4 + 18}$

$a = \frac{10}{22}$

$a = \frac{5}{11}$

Q: What if the equation has multiple variables?

A: If the equation has multiple variables, you need to isolate the variable $a$ on one side of the equation. For example, if the equation is:

$K = 4a + 9ab + 2c$

You can start by subtracting $2c$ from both sides of the equation:

$K - 2c = 4a + 9ab$

Next, you can divide both sides of the equation by $4$:

$\frac{K - 2c}{4} = a + \frac{9ab}{4}$

Finally, you can subtract $\frac{9ab}{4}$ from both sides of the equation:

$\frac{K - 2c}{4} - \frac{9ab}{4} = a$

Therefore, the correct solution to the equation $K = 4a + 9ab + 2c$ for the variable $a$ is:

$a = \frac{K - 2c}{4} - \frac{9ab}{4}$

Q: How do I check if the solution is correct?

A: To check if the solution is correct, you need to substitute the expression for $a$ back into the original equation and check if it holds true. For example, if the solution is:

$a = \frac{K}{4 + 9b}$

You can substitute this expression into the original equation:

$K = 4\left(\frac{K}{4 + 9b}\right) + 9\left(\frac{K}{4 + 9b}\right)b$

Expanding and simplifying the equation, you should get:

$K = K$

This confirms that the solution is correct.

Q: What if I get a different solution?

A: If you get a different solution, it may be due to a mistake in the calculation or a misunderstanding of the equation. You should recheck your work and make sure that you have isolated the variable $a$ correctly. If you are still unsure, you can try to simplify the equation or use a different method to solve for $a$.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation $K = 4a + 9ab$ for the variable $a$. We also discussed how to apply the solution to different scenarios and how to check if the solution is correct. By following these steps, you should be able to solve equations with multiple variables and check if the solution is correct.