In Problems 71-74, use the idea behind Example 15 with a...

In Problems 71-74, use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.
$\left\{\begin{aligned} 25 x+61 y-12 z & =25 \\ 18 x-12 y+7 z & =10 \\ 3 x+4 y-z & =-4\end{aligned}\right.$

Key Concepts

Systems of Linear Equations

This concept refers to a set of equations involving the same set of unknowns, where each equation is linear. In the context of the problem, the system involves three equations with three unknowns, and solving the system means finding a common solution that satisfies all individual equations simultaneously.

Graphical Methods

Graphical methods involve representing each equation as a geometric object, such as a line or a plane, and identifying the point(s) of intersection among them. In three dimensions, each linear equation represents a plane, and the solution to the system corresponds to the intersection of these planes. Using graphing utilities, one can visualize these intersections to approximate solutions.

Graphing Utilities

Graphing utilities are software tools that can plot equations and functions, allowing users to visually inspect and analyze the intersections corresponding to solutions of a system. This approach is especially useful when the system is too complex to solve analytically, as it provides an approximate solution to the equations that can then be refined by other methods if needed.

Numerical Approximation and Rounding

Numerical approximation involves estimating solutions to equations when exact answers are difficult or impossible to obtain. Rounding is an essential part of this process, as it provides a way to express approximate solutions to a specified degree of precision, as required in the problem where answers are rounded to two decimal places.

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