Use a graphing utility to graph the lines in the system. Use...

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\{\begin{array}{r} -\frac{1}{4} x-\frac{1}{2} y=1 \\ 5 x+y=1 \end{array}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically.
$$\left\{\begin{array}{r}
-\frac{1}{4} x-\frac{1}{2} y=1 \\
5 x+y=1
\end{array}\right.$$

Key Concepts

Algebraic Verification

Algebraic methods, such as substitution or elimination, provide a step-by-step process to solve systems of linear equations and confirm solutions obtained graphically. Verifying results algebraically ensures that the solution is accurate and that the values found for the variables satisfy all the equations in the system.

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The goal is to find the values for the variables that satisfy every equation simultaneously. This concept is fundamental in algebra and applies to various fields including engineering, economics, and the sciences.

Graphical Methods for Solving Equations

Graphical methods involve plotting each equation on the coordinate plane and looking for the point or points where the graphs intersect. The intersection represents the solution to the system. This visual approach helps in understanding the relationship between the equations, such as identifying parallel lines or coinciding lines.

Consistent and Inconsistent Systems

A system is called consistent if there is at least one solution that satisfies all equations; it is inconsistent if no such solution exists. In the context of graphing, a consistent system will have graphs that intersect at least once, while an inconsistent system will feature graphs that do not intersect (such as parallel lines).

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