In Problems 19-56, solve each system of equations. If the...

In Problems 19-56, 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.
$$
\left\{\begin{array}{l}
x+y=8 \\
x-y=4
\end{array}\right.
$$

Key Concepts

Consistency and Inconsistency

A system of equations is considered consistent if there is at least one set of values for the variables that satisfies all equations, and inconsistent if no such set exists. Identifying whether a system is consistent is key to understanding the behavior of the lines when graphed; intersecting lines indicate consistency, while parallel lines without intersection indicate inconsistency.

Graphing Linear Equations

Graphing linear equations involves plotting the lines represented by the equations on the coordinate plane. The point where the graphs intersect represents the solution to the system. This visual approach not only confirms algebraic solutions but also provides insight into whether the system has a unique solution, infinitely many solutions, or no solution at all.

System of Linear Equations

A system of linear equations is a set of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. This foundational concept in algebra is used to model and solve problems involving relationships between quantities.

Elimination Method

The elimination method is a technique used to solve systems of equations by strategically adding or subtracting the equations to eliminate one variable, making it easier to solve for the remaining variable(s). This method highlights the structure of linear systems and is particularly useful when the coefficients of variables are easily manipulated to cancel out another variable.

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