If the equations are dependent, write the solution set in...

If the equations are dependent, write the solution set in terms of the variable $z$. (Hint: In Exercises 33-36, let $t=\frac{1}{x}, u=\frac{1}{y},$ and $v=\frac{1}{z} .$ Solve for $t, u,$ and $v,$ and then find \begin{aligned} &2 x+6 y-z=6\\ &4 x-3 y+5 z=-5\\ &6 x+9 y-2 z=11 \end{aligned}

If the equations are dependent, write the solution set in terms of the variable $z$. (Hint: In Exercises 33-36, let $t=\frac{1}{x}, u=\frac{1}{y},$ and $v=\frac{1}{z} .$ Solve for $t, u,$ and $v,$ and then find
\begin{aligned}
&2 x+6 y-z=6\\
&4 x-3 y+5 z=-5\\
&6 x+9 y-2 z=11
\end{aligned}

Key Concepts

Dependent Equations

Dependent equations in a system indicate that one or more of the equations are linear combinations of the others. This means they do not provide new information, leading to infinitely many solutions rather than a unique solution. Recognizing dependency is crucial in determining whether the system is consistent and how to express its general solution.

Parametric Representation

When a system of equations has infinitely many solutions, the solutions are often described in terms of one or more free parameters. This method assigns a parameter (such as z) to the free variable and then expresses the remaining variables in terms of this parameter, making the complete family of solutions clear and manageable.

Substitution Method

The substitution method involves solving one equation for a particular variable and then substituting that expression into the other equations. This method is especially useful in systems where one or more equations are dependent, as it can reduce the number of variables and simplify the process of finding a general solution.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Solving such systems requires techniques such as elimination, substitution, or matrix operations. Understanding these methods is fundamental not only for finding unique solutions but also for recognizing cases where the system has no solution or infinitely many solutions.

Reciprocal Transformation

Reciprocal transformation is a substitution technique where reciprocal values of the original variables are introduced (for example, letting t = 1/x). This transformation can sometimes simplify the relationship between variables in a nonstandard or nonlinear system, ultimately allowing one to apply linear methods and reinterpret the results in terms of the original variables.

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