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. $$\left\{\begin{aligned} x+4 y &=6 \\ \frac{1}{12} x+\frac{1}{3} y &=\frac{1}{2} \end{aligned}\right.$$

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.
$$\left\{\begin{aligned} x+4 y &=6 \\ \frac{1}{12} x+\frac{1}{3} y &=\frac{1}{2} \end{aligned}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. These systems can be solved to find values for the variables that satisfy all the equations simultaneously, and they are fundamental in algebra as they appear in various applications like optimization, economics, and engineering.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation(s). This technique reduces the system to a single equation with one variable, making it easier to solve and back-substitute to find the remaining variables.

Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations such that adding or subtracting them eliminates one of the variables. This yields an equation with fewer variables, which simplifies the process to find the solution of the system.

Classification of Systems (Consistent Independent, Consistent Dependent, Inconsistent)

Systems of linear equations can be classified based on the number of solutions they have: consistent independent systems have a unique solution, consistent dependent systems have infinitely many solutions, and inconsistent systems have no solution. This classification helps in determining the nature of the system before or during the solving process.

Graphical Analysis of Systems

Graphical analysis involves plotting the equations on a coordinate plane to visually inspect their intersections. A unique intersection point indicates a unique solution (consistent independent), overlapping lines indicate infinitely many solutions (consistent dependent), and parallel lines with no intersection indicate no solution (inconsistent).

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