Solve each system by the substitution method. Identify...

Solve each system by the substitution method. Identify inconsistent systems and systems with dependent equations, using set notation to express their solution sets.
$$
\left\{\begin{array}{c}{x-2 y=4} \\ {2 x-4 y=5}\end{array}\right.
$$

Key Concepts

Dependent Systems

Dependent systems are systems of equations that have infinitely many solutions because the equations are essentially multiples of each other. In such cases, the equations represent the same line, and every solution that satisfies one equation will satisfy the other. Dependent systems are identified when manipulation of the equations leads to an identity, such as 0 = 0, which indicates that the system does not uniquely determine the variables.

Solution Set Notation

Solution set notation is a mathematical language used to express the set of all solutions to a system of equations. For a unique solution, the set is typically written as a singleton, for inconsistent systems an empty set, and for dependent systems as a description involving a parameter (or simply the entire line), often using set-builder notation. This notation provides a concise and precise way to represent all possible answers.

Substitution Method

The substitution method is a technique for solving systems of equations where one of the equations is solved for one variable in terms of the other. This expression is then substituted into the other equation, reducing the system to a single equation in one variable. This method is particularly efficient when one equation is easily solvable for a variable, and it is a fundamental tool in algebra for solving linear systems.

Inconsistent Systems

An inconsistent system refers to a set of equations that have no solution because the equations contradict one another. In a graphing context, this typically means the lines represented by the equations are parallel and never intersect. Recognizing an inconsistent system involves finding a contradiction in the algebraic manipulation, such as obtaining a false statement like 0 = 1.

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