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When solving an equation, one of the first steps is often to isolate the variable. This means rearranging the equation so that the variable is alone on one side. Let's start with our example:

\(-\frac{2}{3} c = 18\)

To isolate the variable \(c\), we need to get rid of the fraction \(-\frac{2}{3}\) that is currently multiplying \(c\). Identifying which terms are affecting the variable is key. In this case, \(-\frac{2}{3}\) is the coefficient of \(c\). Once we pinpoint this, we can strategically perform operations to both sides of the equation to isolate \(c\).

By isolating the variable correctly, you help simplify the equation and bring it closer to a solution.

A reciprocal is simply one of a pair of numbers that, when multiplied together, equal 1. For example, the reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\). Now, why do we use reciprocals when solving equations?

Reciprocals are particularly useful when you need to eliminate fractions. Let's go back to our example equation:

\(-\frac{2}{3} c = 18\)

To isolate \(c\), we multiply both sides of the equation by the reciprocal of \(-\frac{2}{3}\), which is \(-\frac{3}{2}\):

\( -\frac{3}{2} \cdot -\frac{2}{3} c = 18 \cdot -\frac{3}{2}\)

This allows the fractions on the left side to cancel out, leaving us with just \(c\). Simplifying the right side gives us the final value \(c = -27\). Always remember to multiply both sides of your equation by the same number to maintain equality.

Prealgebra lays the foundation for all future algebra courses. In prealgebra, you'll work with basic arithmetic, fractions, and simple equations. These skills are critical for solving more complex problems. Let's consider our original example equation again:

\(-\frac{2}{3} c = 18\)

This type of problem typically appears in prealgebra. It involves understanding fractions, reciprocals, and simple multiplication. Key concepts from prealgebra that help in solving such problems include:

• Understanding fractions and how to manipulate them
• Knowing what reciprocals are and how they function in equations
• Basic equation solving skills for isolating variables

Prealgebra provides the essential tools to handle such problems with confidence. Mastering these basics makes algebra and subsequent math subjects much more manageable.