Solving 8/9 = 4x Step-by-Step Guide

Step-by-Step Solution

To solve for $x$ in the equation $\frac{8}{9} = 4x$, we need to isolate $x$ on one side of the equation. This means we want to get $x$ by itself. The equation currently shows that $4x$ is equal to $\frac{8}{9}$. To isolate $x$, we need to undo the multiplication by 4. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 4. Dividing both sides of an equation by the same number maintains the equality, which is a fundamental principle in algebra. This ensures that the value of $x$ we find will indeed satisfy the original equation. So, our next step is to perform this division on both sides of the equation. This process of isolating the variable is at the heart of solving algebraic equations. By carefully applying inverse operations, we can systematically unravel the equation and find the value of the unknown. Let's proceed with the division and see how it simplifies the equation further.

1. Divide Both Sides by 4

To isolate $x$, we divide both sides of the equation $\frac{8}{9} = 4x$ by 4. This gives us:

$\frac{8}{9} \div 4 = 4x \div 4$

Dividing by 4 on both sides ensures that the equation remains balanced. It's like having a weighing scale; if you perform the same operation on both sides, the scale remains balanced. This principle is crucial in algebra and is used extensively in solving equations. When we divide $4x$ by 4, we are left with just $x$ on the right side, which is exactly what we want. On the left side, we have a fraction divided by a whole number, which we need to simplify. Dividing a fraction by a whole number can be thought of as multiplying the fraction by the reciprocal of the whole number. In this case, dividing by 4 is the same as multiplying by $\frac{1}{4}$. This understanding helps us to simplify the left side of the equation effectively. So, let's move on to the next step and perform this division, or rather, multiplication by the reciprocal, to simplify the left side of the equation.

2. Simplify the Left Side

Dividing $\frac{8}{9}$ by 4 is the same as multiplying $\frac{8}{9}$ by $\frac{1}{4}$. So, we have:

$\frac{8}{9} \times \frac{1}{4} = x$

When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. In this case, we multiply 8 by 1 to get the new numerator, and 9 by 4 to get the new denominator. This gives us a new fraction that we can further simplify if possible. Understanding how to multiply fractions is essential for solving many types of algebraic problems. It allows us to combine fractional quantities and express them in a simpler form. In this step, we are essentially combining the fraction $\frac{8}{9}$ with the fraction $\frac{1}{4}$ to find their product, which will give us the value of $x$. Let's perform the multiplication and see what fraction we obtain. This will bring us closer to the final simplified value of $x$. Multiplying fractions is a straightforward process, and with practice, it becomes second nature.

$\frac{8 \times 1}{9 \times 4} = x$

$\frac{8}{36} = x$

Now we have $\frac{8}{36} = x$. However, this fraction can be simplified further. Both the numerator (8) and the denominator (36) have common factors, which means we can divide both by the same number to get a simpler fraction. Simplifying fractions is important because it allows us to express the value of $x$ in its most basic form. A simplified fraction is easier to understand and work with in further calculations. To simplify $\frac{8}{36}$, we need to find the greatest common factor (GCF) of 8 and 36 and divide both the numerator and the denominator by that GCF. The GCF is the largest number that divides both 8 and 36 without leaving a remainder. Let's identify the GCF and then simplify the fraction to find the value of $x$ in its simplest form. Simplifying fractions not only makes the answer cleaner but also demonstrates a good understanding of mathematical principles.

3. Simplify the Fraction

The fraction $\frac{8}{36}$ can be simplified. Both 8 and 36 are divisible by 4. Dividing both the numerator and the denominator by 4, we get:

$\frac{8 \div 4}{36 \div 4} = x$

$\frac{2}{9} = x$

So, the simplified value of $x$ is $\frac{2}{9}$. This is the simplest form of the fraction, as 2 and 9 have no common factors other than 1. We have now successfully isolated $x$ and found its value. This process demonstrates the importance of simplifying fractions to obtain the most concise and clear answer. Simplifying is a crucial step in solving algebraic equations, as it ensures that the answer is expressed in its most manageable form. In this case, $\frac{2}{9}$ is easier to work with than $\frac{8}{36}$. Therefore, we have completed the process of solving for $x$ and simplifying the result. The final step is to state the solution clearly.

Final Answer

Therefore, $x = \frac{2}{9}$. This is the solution to the equation $\frac{8}{9} = 4x$, simplified as much as possible. We have successfully isolated $x$ and expressed its value as a simple fraction. This result can be verified by substituting $\frac{2}{9}$ back into the original equation and checking if it holds true. If we substitute $x = \frac{2}{9}$ into $\frac{8}{9} = 4x$, we get $\frac{8}{9} = 4 \times \frac{2}{9}$, which simplifies to $\frac{8}{9} = \frac{8}{9}$. This confirms that our solution is correct. The ability to verify solutions is an important skill in mathematics, as it allows us to check our work and ensure accuracy. In this case, we have not only solved the equation but also verified our solution, giving us confidence in our answer. So, we can confidently state that $x = \frac{2}{9}$ is the solution to the given equation.

Conclusion

In conclusion, solving for $x$ in the equation $\frac{8}{9} = 4x$ involves dividing both sides by 4 and then simplifying the resulting fraction. The key steps are: dividing both sides by 4, simplifying the left side by multiplying fractions, and reducing the fraction to its simplest form. This process demonstrates fundamental algebraic principles, such as using inverse operations to isolate variables and simplifying fractions. Mastering these skills is crucial for solving more complex algebraic problems. Remember, practice is key to improving your problem-solving abilities. By working through various examples and understanding the underlying concepts, you can build your confidence and competence in algebra. This example provides a solid foundation for further learning and exploration in mathematics. Keep practicing and applying these techniques, and you'll be well-equipped to tackle a wide range of algebraic challenges. We hope this guide has been helpful in clarifying the process of solving for $x$ in this equation. If you have any further questions or need more examples, don't hesitate to explore additional resources and seek assistance.