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

Solve each system of equations. If the system has no solution, state that it is inconsistent.
$$
\left\{\begin{array}{l}
\frac{1}{3} x-\frac{3}{2} y=-5 \\
\frac{3}{4} x+\frac{1}{3} y=11
\end{array}\right.
$$

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Key Concepts

Consistent and Inconsistent Systems

A consistent system is one that has at least one solution, meaning the equations intersect at one or more points. Conversely, an inconsistent system has no solution, meaning the equations do not intersect. Recognizing whether a system is consistent or inconsistent is important as it informs the approach to finding solutions or concluding that none exist.

Elimination Method

The elimination method is a systematic technique used to solve systems of equations by adding or subtracting the equations to cancel one of the variables. This method often involves scaling the equations so that the coefficients of one of the variables match, thereby simplifying the system to make it easier to solve.

Systems 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 values of these variables that satisfy all the equations simultaneously. This concept is central to algebra and has widespread applications in fields such as engineering, economics, and physical sciences.

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