91Ó°ÊÓ

When solving equations that include fractions, the goal is to find the value of the variable. Fractions can make equations tricky, but breaking the problem into smaller steps helps. First, make sure to write down the equation clearly. In this exercise, we started with \(\frac{3}{4} - \frac{1}{x} = \frac{7}{8}\). Next, move terms involving the variable to one side to isolate them. Then, work on simplifying the fractions. It's often helpful to have a common denominator when adding or subtracting fractions. Always take care to perform the same operation on both sides of the equation to keep it balanced. Eventually, simplify to a form where you can see the solution clearly. Remember, practice is key to becoming comfortable with fraction equations.

Isolating the variable means getting the variable alone on one side of the equation. This helps in solving the equation effectively. In this example, our variable is \(x\). Start by moving terms that do not include the variable to the opposite side. To isolate \(x\), we first moved \( \frac{1}{x} \) to the right-hand side: \( \frac{3}{4} = \frac{7}{8} + \frac{1}{x} \). Then, we subtracted \( \frac{7}{8} \) from both sides to further isolate \( \frac{1}{x}\): \( \frac{3}{4} - \frac{7}{8} = \frac{1}{x} \). The next step involved simplifying the fractions, making sure to handle each part carefully. Once simplified, we had \( -\frac{1}{8} = \frac{1}{x} \).

The reciprocal of a number is 1 divided by that number. It essentially 'flips' the fraction. For instance, the reciprocal of \( 2 \) is \( \frac{1}{2}\), and the reciprocal of \(\frac{3}{4}\) is \( \frac{4}{3} \). In our example, we had \( -\frac{1}{8} = \frac{1}{x} \). To solve for \( x \), we need to take the reciprocal of both sides. The reciprocal of \( -\frac{1}{8} \) is \(-8\), thus giving us \( x = -8 \). This process is crucial for solving equations where the variable is part of a fraction. Taking the reciprocal turns the fraction involving the variable into a simpler equation, making it straightforward to find the value of the variable.