Solve the system. Then classify the system as consistent and...

Solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.
$\begin{aligned} & \frac{1}{2} x-3 y=10 \\ & \frac{1}{4} x+2 y=-2\end{aligned}$

Key Concepts

Elimination Method

The elimination method is a common technique used to solve systems of linear equations. By adding or subtracting the equations, one variable is eliminated, thereby reducing the system to a simpler one that is easier to solve. This method is particularly useful in identifying the structure and consistency of the system.

Inconsistent Systems

A system is inconsistent if there is no solution to the system. This occurs when the equations represent parallel lines (or hyperplanes) that never intersect, meaning there is no common set of variable values that satisfies all the equations.

Consistent and Independent Systems

A system is consistent and independent if it has exactly one unique solution. This situation occurs when the equations represent lines that intersect at exactly one point, meaning there is one set of variable values that works for all equations.

System of Linear Equations

A system of linear equations is a set of two or more equations that involve the same variables, where each equation represents a line (in two dimensions) or a hyperplane (in higher dimensions). The goal is to find the values of the variables that satisfy all the equations simultaneously.

Consistent and Dependent Systems

A system is consistent and dependent when it has infinitely many solutions. This happens if all equations in the system describe the same line (or plane), so every solution to one equation is also a solution to the others.

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