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.
$\left\{\begin{aligned} 25 x+61 y-12 z & =10 \\ 18 x-12 y+7 y & =-9 \\ 3 x+4 y-z & =12\end{aligned}\right.$

Key Concepts

Rounding and Numerical Approximation

Rounding is the process of adjusting numerical values to a desired degree of precision, which in this context is to two decimal places. Numerical approximation, often achieved through graphing utilities, provides a practical approach to finding solutions when exact values are hard to determine. This ensures that the answers are presented in a concise and usable form while balancing precision and clarity.

Use of Graphing Calculators

Graphing calculators or utility software enable users to plot equations accurately and find intersections more efficiently. These tools are especially helpful when manual graphing is complex or when the equations involve multiple variables. By leveraging these utilities, one can obtain numerical approximations for the solutions of the system, which is useful in cases where exact analytical solutions are difficult to compute.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. The goal is to find a set of values for the variables that satisfies every equation in the system simultaneously. These systems can be solved via various methods such as substitution, elimination, matrix operations, or by using graphical methods.

Graphical Solution Methods

Graphical methods involve plotting the graphs corresponding to each equation and identifying their point(s) of intersection. The intersection represents the solution to the system since it is the common set of variable values that satisfies all the equations in the system. This approach is particularly useful for visualizing the behavior and relationship of the equations.

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