Solve using the elimination method. Also determine whether...

Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent. Use a graphing calculator to check your answer.
$$\begin{aligned}
&3 x-12 y=6\\
&2 x-8 y=4
\end{aligned}$$

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Key Concepts

Elimination Method

The elimination method is an algebraic technique used to solve systems of equations by adding or subtracting the equations to eliminate one of the variables. This reduction simplifies the system to a single equation with one variable, making it easier to solve.

System of Linear Equations

A system of linear equations is a collection of two or more linear equations with the same set of variables. The solution of the system is the set of variable values that satisfy every equation simultaneously.

Consistent and Inconsistent Systems

A consistent system is one that has at least one solution, whether unique or infinitely many. An inconsistent system, on the other hand, has no solution because the equations contradict each other.

Dependent and Independent Equations

Dependent equations are those that are multiples of each other, resulting in infinitely many solutions as they represent the same line. Independent equations intersect at a single unique point, yielding exactly one solution.

Graphical Verification

Graphical verification involves using tools like graphing calculators to plot the equations. This visual method helps confirm the solution by showing whether the lines intersect at a point, are coincident, or do not intersect at all.

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