Let's dive into solving for x in the equation (1/f) = (1/6) + (1/7) - (x/8). This might seem daunting at first, but don't worry, we'll break it down into manageable steps. Our goal is to isolate x on one side of the equation, which means we need to get rid of all the other terms around it. To make things easier, we'll first focus on simplifying the right side of the equation by combining the fractions.

Combining Fractions

The first step in solving for x involves simplifying the right-hand side of the equation. We have (1/6) + (1/7). To add these fractions, we need a common denominator. The least common multiple of 6 and 7 is 42. So, we convert each fraction to have this denominator:

(1/6) = (1 * 7) / (6 * 7) = 7/42 (1/7) = (1 * 6) / (7 * 6) = 6/42

Now we can add them together:

(7/42) + (6/42) = (7 + 6) / 42 = 13/42

So, the right side of the equation now looks like this: (13/42) - (x/8). Remember, our original equation was (1/f) = (1/6) + (1/7) - (x/8). We've simplified it to (1/f) = (13/42) - (x/8). This is progress! We're one step closer to isolating x. Keep pushing forward; you're doing great! The key here is to take things one step at a time and not get overwhelmed by the complexity. Fraction manipulation can be tricky, but with practice, it becomes second nature.

Isolating the Term with x

Next, we want to isolate the term containing x, which is -(x/8). To do this, we need to get rid of the (13/42) term on the right side of the equation. We can achieve this by subtracting (13/42) from both sides of the equation. This maintains the balance of the equation:

(1/f) - (13/42) = (13/42) - (x/8) - (13/42)

This simplifies to:

(1/f) - (13/42) = -(x/8)

Now, we have the term with x isolated on one side. To proceed further, we need to deal with the (1/f) term. Since the value of f is not provided, we'll keep it as a variable and continue solving for x in terms of f. If we had a specific value for f, we could substitute it in and get a numerical answer for x. But for now, we'll keep it general. Remember, algebra is all about manipulating symbols and equations to find relationships between variables.

Solving for x

We have the equation (1/f) - (13/42) = -(x/8). To solve for x, we need to get rid of the -8 in the denominator. We can do this by multiplying both sides of the equation by -8:

-8 * [(1/f) - (13/42)] = -8 * [-(x/8)]

Distribute the -8 on the left side:

(-8/f) + (8 * 13/42) = x

Simplify the second term on the left side:

(8 * 13) / 42 = 104 / 42 = 52 / 21

So, the equation becomes:

(-8/f) + (52/21) = x

Therefore, x = (-8/f) + (52/21). We have now solved for x in terms of f. If you have a specific value for f, you can plug it into this equation to find the numerical value of x. This is a crucial point: the solution for x depends on the value of f. Without knowing f, we can only express x in terms of f. * Keep practicing, and you'll become a master of algebraic manipulation!*

Final Answer

The final solution for x, in terms of f, is:

x = (-8/f) + (52/21)

If you have a value for f, substitute it into the equation to find the corresponding value for x. Otherwise, this is the most simplified form of the solution. Remember to double-check your work by plugging the value of x (in terms of f) back into the original equation to ensure it holds true. This is a good habit to develop in algebra – always verify your solutions! Guys, you've successfully navigated this algebraic problem! Give yourself a pat on the back. This is a great example of how to break down a complex equation into smaller, more manageable steps.

Example with f = 1

Let's say f = 1. We can plug this value into our solution for x:

x = (-8/1) + (52/21)

x = -8 + (52/21)

To combine these terms, we need a common denominator, which is 21:

x = (-8 * 21/21) + (52/21)

x = (-168/21) + (52/21)

x = (-168 + 52) / 21

x = -116 / 21

So, if f = 1, then x = -116/21. This demonstrates how the value of f directly impacts the value of x. Always remember to consider the relationships between variables when solving equations. The example illustrates that based on the value of f we can find x value. Don't be afraid to explore different values of f to see how they affect x. This can help you develop a deeper understanding of the equation.

Verification

Let’s verify our solution when f = 1 and x = -116/21. The original equation is (1/f) = (1/6) + (1/7) - (x/8). Substituting the values:

(1/1) = (1/6) + (1/7) - (-116/21) / 8

1 = (1/6) + (1/7) + (116/21) / 8

1 = (1/6) + (1/7) + (116/21 * 1/8)

1 = (1/6) + (1/7) + (116/168)

1 = (1/6) + (1/7) + (29/42)

Now, find a common denominator, which is 42:

1 = (7/42) + (6/42) + (29/42)

1 = (7 + 6 + 29) / 42

1 = 42 / 42

1 = 1

The equation holds true. Therefore, our solution is correct. Verification is a critical step in problem-solving. It gives you confidence in your answer and helps you catch any potential errors. You see here that verifying an equation ensures that the equation holds true or not. So it's a clever practice to use in math problems. You've successfully verified the equation which ultimately gives you the peace of mind!

Further Exploration

To enhance your understanding, try varying the value of f and observing how x changes. This exercise will help you grasp the relationship between f and x more intuitively. You can also explore similar equations with different combinations of fractions and variables to strengthen your problem-solving skills. Practice makes perfect, so keep challenging yourself with new and diverse algebraic problems.

Remember, the key to mastering algebra is to break down complex problems into smaller, more manageable steps, and to always verify your solutions. With consistent effort and practice, you can conquer any equation that comes your way! Guys, you are now equipped with the knowledge to solve such questions and I hope you will come back for more! Happy solving!