Use the idea behind Example 15 with a graphing utility to...

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$$
\left\{\begin{aligned} 25 x+61 y-12 z &=15 \\ 18 x-12 y+7 z &=-3 \\ 3 x+4 y-& z=12 \end{aligned}\right.
$$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The solution to the system is the set of variable values that simultaneously satisfy all equations. In the context of multiple variables, each equation represents a geometric object (such as a line or plane) and the solution corresponds to their point (or points) of intersection.

Graphical Methods and Utility Tools

Graphical methods involve representing equations on a coordinate system to visualize their solutions. Graphing utilities allow users to accurately plot equations and identify the points of intersection. This approach is especially useful when solving systems with multiple equations as it provides a visual means to estimate and verify the solutions.

Intersection of Graphs

The intersection of graphs in the context of systems of equations refers to the point or region where the graphs of the individual equations overlap. This common intersection represents the set of variable values that satisfy all given equations simultaneously. In higher dimensions, like with three variables, these intersections can correspond to the meeting point of planes.

Rounding and Approximation

Rounding and approximation are essential when working with numerical solutions, especially when the precise solution involves decimals or irrational numbers. It is common practice to round the results to a specified number of decimal places for clarity and practical use, ensuring that the solution is both accurate and easily interpretable.

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