In Exercises 1-8, take a trip down memory lane and solve the...

In Exercises $1-8$, 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. $$ \left\{\begin{array}{r} x+4 y=6 \\ \frac{1}{12} x+\frac{1}{3} y=\frac{1}{2} \end{array}\right. $$

In Exercises $1-8$, 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.
$$
\left\{\begin{array}{r}
x+4 y=6 \\
\frac{1}{12} x+\frac{1}{3} y=\frac{1}{2}
\end{array}\right.
$$

Key Concepts

Graphical Representation

Graphical methods involve plotting each equation on a coordinate plane as a line. The point where the lines intersect represents the solution to the system. This visualization helps to confirm the algebraic solution and classify the system by showing whether the lines coincide, intersect at a single point, or are parallel.

Classification of Systems

Systems of linear equations are classified based on the number and type of solutions. A consistent independent system has exactly one unique solution where the lines intersect at one point. A consistent dependent system has infinitely many solutions, meaning the equations represent the same line. An inconsistent system has no solutions because the equations represent parallel lines that never intersect.

Elimination Method

In the elimination method, the objective is to eliminate one variable by adding or subtracting the equations from each other. By aligning the coefficients of one variable, you can cancel it out, reducing the system to a simpler equation. This technique is efficient for systems with easily manipulable coefficients.

Substitution Method

This method involves solving one of the equations for one variable and then substituting that expression into the other equation(s). It reduces the number of variables in an equation, making it easier to find the solution step by step. It is particularly useful when one equation is already or can be easily solved for one variable.

System of Linear Equations

These are sets of two or more equations involving the same set of variables where the goal is to find a common solution that satisfies all equations simultaneously. They are foundational in algebra and applied mathematics, serving as models for real-world scenarios where multiple conditions must be met at once.

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