Isolation of a variable

Functions of several variables: Basic concepts and visualization

Isolation of a variable

Rewriting implicit relationship between three or more variables in a form in which one of the variables, say $v$ , is present on its own on the left-hand side of an equation, i.e., creating an equation of the form $v = \mathrm{formule\;without\;}v$ , is called the isolaton of the variable \small.\) You get then a definition of a function of several variables The example below shows how it works.

In the relationship $\displaystyle y=\frac{7x+8z}{-6x+11z}$ you immediately see that $y$ is a function of $x$ and $z$ .
But is $z$ also a function of $x$ and $y$ ? And, if so, what is the function definition?
In other words, can you express $z$ in $x$ and $y$ , in the form $z=\mathrm{formule\;in\;} x\mathrm{\;and\;}y$ .

You can also reach the solution by increments entering in the form of equations:
Then you see if you are still on track, but in the end you must get to the equation in the form $z=\ldots$ .

You try to isolate the variable $z$ te isoleren in the given relation $\displaystyle y=\frac{7x+8z}{-6x+11z}$ .
Multiply both sides of the equation with $-6x+11z$ and simplify:

$\begin{aligned} (-6x+11z)y &= \frac{(7x+8z)(-6x+11z)}{(-6x+11z)}\\ \\ -6xy+11yz&=7x+8z \end{aligned}$ Now you have obtained an equation without denominators and subsequently only have to manipulate polynomial equations. Move all terms with $z$ to the left-hand side and move all terms without $z$ to the right-hand side, and factorize:

$\begin{aligned}11yz-8z &= 6xy+7x\\ \\ z(11y-8) &=x(6y+7) \end{aligned}$ Division of the last equation by $11y-8$ leads to the requested form:

$z=\frac{x(6y+7)}{11y-8}$ This can be read as definiton of a function $z$ of $x$ and $y.$

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