91Ó°ÊÓ

When solving linear equations, understanding how to combine like terms is critical. Like terms are terms that have the same variable raised to the same power. In our problem, \(\frac{3}{4} x\) and \(\frac{1}{2} x\) are like terms since both have the variable \(x\). To combine them, you need a common denominator so you can easily add the fractions together. In our problem, the common denominator for \(\frac{3}{4}\) and \(\frac{1}{2}\) is 4. We convert \(\frac{1}{2}\) to \(\frac{2}{4}\) in order to combine them nicely into \(\frac{5}{4} x\). This simplifies our equation and sets the stage for isolating the variable.

Understanding the concept of a common denominator is essential when working with fractions. A common denominator is a shared multiple of the denominators of two or more fractions, which allows you to combine the fractions easily. For example, in our exercise, we had the fractions \(\frac{3}{4}\) and \(\frac{1}{2}\). The least common denominator between 4 and 2 is 4. This step involves converting each fraction to an equivalent fraction with the same denominator, making it easier to perform addition or subtraction. Here, \(\frac{1}{2} x\) is rewritten as \(\frac{2}{4} x\), allowing us to sum them up as \(\frac{5}{4} x\).

The concept of reciprocals is pivotal in solving equations involving fractions. A reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\). When you multiply a number by its reciprocal, the result is always 1. In our equation, after combining like terms, we reached \(\frac{5}{4} x = -10\). The goal is to isolate \(x\). To do this, we multiply both sides of the equation by the reciprocal of \(\frac{5}{4}\), which is \(\frac{4}{5}\). Therefore, \(x = -10 \times \frac{4}{5}\), and simplifying this multiplication yields \(x = -8\). Reciprocals provide a straightforward way to manage and solve equations involving fractions.