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In algebra, solving for a specific variable means maneuvering an equation until that variable is on one side of the equal sign, and everything else is on the other. This crucial step is known as 'isolating the variable.' To do this, you have to perform a series of operations that reverse the ones initially applied to the variable. Let's take the exercise example \(v = \frac{3}{4} w\). In this equation, our goal is to isolate \(w\).

We begin by looking at the equation and deciding what operations are currently being applied to \(w\), which in our case is multiplication by \(\frac{3}{4}\). To reverse this operation, we would need to do the opposite, which leads us to our next step: multiplying by the reciprocal.

Multiplication by the reciprocal is a method used to eliminate fractions from an equation or to isolate a variable. The reciprocal of a number is what you multiply the original number by to get 1. For a fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

In the given exercise, we intend to remove the fraction \(\frac{3}{4}\) that's connected to \(w\) by multiplying it by its reciprocal, \(\frac{4}{3}\). So, we perform the same operation to both sides of the equation: \(\frac{4}{3}v = \left(\frac{4}{3} \times \frac{3}{4}\right)w\). Since multiplying a number by its reciprocal results in 1, the equation simplifies to \(w = \frac{4}{3}v\) effectively isolating the variable \(w\).

Simplifying equations is a fundamental aspect of solving algebraic expressions. It involves reducing an equation to its simplest form, making the solution clear and manageable. This often includes combining like terms, eliminating fractions, and reducing expressions to lowest terms.

In our step by step solution, after multiplying by the reciprocal, we simplify the equation. The multiplication of \(\frac{4}{3}\) by \(\frac{3}{4}\) can be simplified by canceling out the common factors in the numerator (top number) and the denominator (bottom number). Here, 3 in the numerator and the denominator cancels out as well as the 4, leaving us with \(w = \frac{4}{3}v\), the simplest form of the equation where \(w\) is alone on one side. When you simplify equations, always ensure that the final result presents the clearest path to the solution, which in the context of a linear equation is typically a variable isolated on one side.