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One of the fundamental techniques in solving linear equations is isolating the variable. This means manipulating the equation so that the variable you want to solve for is on one side of the equation by itself. In the given exercise, the goal is to solve for a when the equation is \( \frac{5}{6} a = -10 \).

To isolate \( a \), we need to undo the operation that is being applied to it. In this case, \( a \) is being multiplied by \( \frac{5}{6} \), so the opposite operation—division—will help us isolate \( a \). It's akin to untangling something; by applying the opposite operation, we 'untangle' \( a \) from the numbers it is combined with, making it stand alone on one side of the equation.

Remember, whatever action is taken to one side of the equation must also be applied to the other to maintain balance, as the equality of the equation has to be preserved. This is a core principle in algebra: the equation must remain balanced at all times for the solution to be correct.

The division of fractions is another crucial concept in algebra, especially when working with equations like \( \frac{5}{6} a = -10 \). If you need to divide by a fraction, like \( \frac{5}{6} \), the process actually involves multiplying by its reciprocal. The reciprocal of a fraction is simply created by flipping the numerator and the denominator.

Therefore, when dividing by \( \frac{5}{6} \), you multiply by its reciprocal, which is \( \frac{6}{5} \). In practical terms, to divide \( -10 \) by \( \frac{5}{6} \: \)

• Flip \( \frac{5}{6} \) to get its reciprocal \( \frac{6}{5} \).
• Multiply \( -10 \) by \( \frac{6}{5} \) to find the value of \( a \).

This process simplifies the operation and enables you to solve the equation without dealing with nested fractions.

The goal of equation simplification is to make the equation as straightforward as possible to solve. During this step, unnecessary complexities are removed, and the equation is broken down into its simplest form. After dividing by \( \frac{5}{6} \), we have \( a = \frac{-10}{\frac{5}{6}} \). By simplifying, we multiply \( -10 \) by the reciprocal of \( \frac{5}{6} \), resulting in \( a = -12 \).

Simplification often involves performing arithmetic operations such as addition, subtraction, multiplication, and division. It may also include combining like terms or factoring. In this exercise, simplifying makes it clear that the solution for \( a \) is \( -12 \: \), providing a straightforward answer that can be easily checked for correctness by substituting \( a \) back into the original equation.