Take a trip down memory lane and solve the given system...

Take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$\left\{\begin{aligned} \frac{1}{2} x-\frac{1}{3} y &=-1 \\ 2 y-3 x &=6 \end{aligned}\right.$$

Take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$\left\{\begin{aligned} \frac{1}{2} x-\frac{1}{3} y &=-1 \\ 2 y-3 x &=6 \end{aligned}\right.$$

Key Concepts

Algebraic and Graphical Verification

Algebraic verification involves solving the system using methods like substitution or elimination and then checking that the solution satisfies all original equations, while graphical verification involves plotting the equations on a coordinate plane to visually confirm where they intersect (or fail to intersect). Using both approaches provides a robust check on the solution and offers insight into the system's geometric interpretation.

Consistency Classification

The classification of systems of linear equations determines whether they have a unique solution, infinitely many solutions, or no solution. A consistent independent system has exactly one solution, a consistent dependent system has infinitely many solutions (indicating that the equations describe the same line), and an inconsistent system has no solution because the lines are parallel. This classification is key in understanding the nature of the solution set of the system.

Elimination Method

The elimination method involves adding or subtracting equations (or sometimes multiplying them by appropriate numbers) in order to eliminate one of the variables. This technique is useful when coefficients of a variable align or can be made to align, allowing the system to be reduced to a single equation in one variable. It is particularly effective for systems of two equations.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the remaining equations. This method is especially useful when one of the equations is already solved for a specific variable or can be easily manipulated to isolate a variable, thereby reducing the system to fewer equations with fewer unknowns.

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 values for the variables that satisfy all equations simultaneously. These systems can be solved using various methods and are fundamental to algebra and applied mathematics, appearing in contexts such as engineering, economics, and science.

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