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
$$\left\{\begin{array}{c}
25 x+61 y-12 z=15 \\
18 x-12 y+7 z=-3 \\
3 x+4 y-z=12
\end{array}\right.$$

Key Concepts

Numerical Approximation and Rounding

Numerical approximation involves finding approximate solutions when exact values may not be practical or possible to obtain. Rounding these approximations to a specified number of decimal places, such as two decimal places in many practical problems, helps simplify and standardize the presentation of results while maintaining a desired level of precision.

Intersection of Planes

In three-dimensional space, each linear equation represents a plane. The solution to a system of three linear equations is the point or line where these planes intersect. Analyzing the intersections graphically can provide insights into whether the system has a unique solution, infinitely many solutions, or no solution at all.

Graphing Utilities

Graphing utilities are tools, often software-based, that allow users to visualize equations and their intersections. In the context of systems of equations, these tools enable one to plot multiple equations—typically as lines in two dimensions or planes in three dimensions—and visually identify the point(s) of intersection, which correspond to the solution(s) of the system.

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In the context of solving these systems, the goal is to find the values for the variables that satisfy all equations simultaneously. Methods for solving such systems include substitution, elimination, matrix operations, and using graphing utilities.

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