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.
$$
\left\{\begin{array}{l}
\frac{1}{2} x+\frac{1}{3} y=3 \\
\frac{1}{4} x-\frac{2}{3} y=-1
\end{array}\right.
$$

Key Concepts

Graphical Representation of Solutions

Graphing the equations of a system visually represents the relationships between the variables. Each linear equation corresponds to a line in a coordinate plane, and the point(s) where these lines intersect represents the solution(s) to the system. This method provides an intuitive understanding of whether the system is consistent (intersecting lines) or inconsistent (parallel lines with no intersection).

Consistency of Systems

The concept of consistency in systems of equations involves determining whether there exists a solution that satisfies all equations. A consistent system has at least one solution (which may be unique or infinitely many), while an inconsistent system has no solution. Recognizing whether a system is consistent is crucial in understanding the nature of the solutions being sought.

Elimination Method

The elimination method involves combining the equations in a way that cancels out one of the variables, making it easier to solve for the remaining variable. This is typically done by multiplying the equations by suitable constants, and then adding or subtracting them to eliminate one of the unknowns. Once one variable is found, substitution into one of the original equations can be used to determine the other variable.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find the values of these variables that satisfy all equations simultaneously. This type of problem is a fundamental part of algebra and is used to model and solve real-world situations where multiple conditions must be met at once.

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