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Solving algebraic fractions involves finding the value of the unknown variable that makes the fraction statements true. When presented with an equation like \(\frac{x}{9}=\frac{-45}{27}\), the first step is often to eliminate the fractions to simplify the equation.

To accomplish this, we use a method called cross multiplication, where we multiply the numerator of one fraction by the denominator of the other and vice versa. This technique relies on the principle of the cross product being equal for proportions — that is, if \(\frac{a}{b} = \frac{c}{d}\), then \(a * d = b * c\). After cross multiplying, we obtain a simpler linear equation without fractions, making it easier to solve for the variable.

The goal of isolating variables is to rewrite equations so the variable we're solving for is on one side of the equation by itself. In the exercise \(\frac{x}{9}=\frac{-45}{27}\), after cross multiplication we get \(27x = -405\). Here, 'x' is still not isolated.

To isolate 'x', we perform operations that will 'undo' anything that is currently being done to 'x'. Since 'x' is being multiplied by 27, we do the opposite operation — division. We divide both sides of the equation by 27: \(\frac{27x}{27} = \frac{-405}{27}\), which simplifies to \(x = -15\). The variable 'x' is now isolated on one side of the equation, giving us the solution.

Simplifying equations makes them easier to solve by eliminating unnecessary complexity. This involves reducing fractions, combining like terms, and removing parentheses whenever possible. In the given problem, after isolating the variable 'x', we simplify the equation by dividing -405 by 27 to get \(x = -15\).

Simplification steps are critical in making sure the solutions found are in their simplest form, a practice which minimizes mistakes and confusion. It’s an essential skill in algebra that not only helps to solve textbook problems but also has practical applications in science, engineering, economics, and various other fields where mathematical models are used.