Solve each system by substitution. Determine whether each...

Solve each system by substitution. Determine whether each system is independent, inconsistent, or dependent.
$$
\begin{array}{l}
\frac{1}{2} x-\frac{1}{3} y=12 \\
\frac{1}{4} x-\frac{1}{2} y=1
\end{array}
$$

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Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that are solved together to find a common solution for the variables. The goal is to determine values for the variables that make all the equations true simultaneously.

Substitution Method

The substitution method involves solving one of the equations for one variable and substituting that expression into the other equation(s). This reduces the system to a single equation with one variable, which can then be solved to find the value of the substituted variable.

Independent System

An independent system is one that has exactly one unique solution. This situation arises when the graphs of the equations intersect at a single point, meaning the constraints imposed by each equation are distinct.

Dependent System

A dependent system is one that has infinitely many solutions. This occurs when the equations are essentially the same, meaning one equation can be derived from the other through algebraic manipulation, so they represent the same line.

Inconsistent System

An inconsistent system is one that has no solutions. This happens when the graphs of the equations are parallel and do not intersect, indicating that the system's equations impose conflicting constraints.

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