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The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and then substituting this expression into another equation. This method allows us to reduce a system of multiple equations into a single equation with one variable.

To use the substitution method effectively, select the equation that can be most easily manipulated to express one variable in terms of the other. For example, in our given exercise with equations \(x^{2}+y^{2}-9=0\) and \(2x-y-3=0\), we choose the latter for it can be rearranged to \(y=2x-3\), a simpler form in terms of \(x\).

The next step is substituting this expression for \(y\) into the other equation. This elimination of \(y\) leads to an equation solely in terms of \(x\), which can be solved using standard algebraic methods. Once \(x\) is found, substitute it back into either of the original equations to find the corresponding value of \(y\).

A system of equations consists of two or more equations with multiple variables that are solved simultaneously. These systems can be linear or nonlinear, and the solutions are pairs or sets of values that satisfy all equations in the system simultaneously.

In our exercise, we have a nonlinear system because at least one equation includes variables to a power other than one, \(x^{2}\) and \(y^{2}\). When solving a nonlinear system using the substitution method, the process involves isolating one variable, substituting it into other equations, and solving for the remaining variables sequentially.

It is crucial to find values that satisfy both equations since a legitimate solution must align with the entire system. For instance, if an obtained value of \(x\) does not produce a corresponding \(y\) that satisfies both original equations, then it cannot be considered a valid solution.

Algebraic solutions pertain to finding answers to equations using algebraic manipulations. These manipulations can include operations such as factoring, expanding, and simplifying expressions.

Building from the example given, after substituting \(y\) with \(2x-3\) into the equation \(x^{2}+y^{2}-9=0\), we obtained \(5x^{2}-12x=0\). To solve this, we can use factoring, a method where we express an equation as a product of its factors. After finding the factored form \(x(5x-12)=0\), we apply the zero product property.

With algebraic solutions, it’s imperative to consider all possible values resulting from the factoring process. With the equation \(x(5x-12)=0\), we identify the solutions for \(x\) as \(x=0\) and \(x=\frac{12}{5}=2.4\). Once we've determined the values of \(x\), we can then find the corresponding \(y\) values to complete the algebraic solution to the system of equations.