Solve each system of equations. If the system has no...

Solve each system of equations. If the system has no solution, say that it is inconsistent. For Problems 19-30, graph the lines of the system.
$$
\left\{\begin{array}{r}
\frac{1}{2} x+y=-2 \\
x-2 y=8
\end{array}\right.
$$

Key Concepts

Consistency and Inconsistency

The concept of consistency in a system of equations refers to whether the system has at least one solution. A consistent system either has one unique solution or infinitely many solutions, while an inconsistent system has no solutions due to the express representation of parallel lines or contradictory equations.

Graphing Linear Equations

Graphing involves plotting each linear equation on a coordinate plane and finding their point of intersection. This visual strategy not only helps in verifying the solution from algebraic methods but also provides insight into the nature of the system, such as whether the lines intersect at a point (indicating a unique solution), coincide (indicating infinitely many solutions), or are parallel (indicating an inconsistent system with no solution).

Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables. This method simplifies the process of finding the variable values by reducing the system to a simpler equation with one variable, which is particularly useful when dealing with coefficients or fractions.

System of Linear Equations

A system of linear equations is a collection of one or more equations made up of the same set of variables. Solving such a system means finding values for the variables that make all the equations true at the same time. These systems can have a unique solution, infinitely many solutions, or no solution at all, which is crucial for determining the relationships between the lines or planes they represent.

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