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 \(\displaystyle y=\frac{13x+2z}{-9x+7z}\) 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 \(z=\ldots\).
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 \(\displaystyle y=\frac{13x+2z}{-9x+7z} \).
Multiply both sides of the equation with \(-9x+7z\) and simplify:
\[\begin{aligned}
(-9x+7z)y &= \frac{(13x+2z)(-9x+7z)}{(-9x+7z)}\\ \\
-9xy+7yz&=13x+2z
\end{aligned}\] 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: \[\begin{aligned}7yz-2z &= 9xy+13x\\ \\
z(7y-2) &=x(9y+13)
\end{aligned}\] Division of the last equation by \(7y-2\) leads to the requested form: \[z=\frac{x(9y+13)}{7y-2}\] This can be read as definiton of a function \(z\) of \(x\) and \(y.\)
Multiply both sides of the equation with \(-9x+7z\) and simplify:
\[\begin{aligned}
(-9x+7z)y &= \frac{(13x+2z)(-9x+7z)}{(-9x+7z)}\\ \\
-9xy+7yz&=13x+2z
\end{aligned}\] 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: \[\begin{aligned}7yz-2z &= 9xy+13x\\ \\
z(7y-2) &=x(9y+13)
\end{aligned}\] Division of the last equation by \(7y-2\) leads to the requested form: \[z=\frac{x(9y+13)}{7y-2}\] This can be read as definiton of a function \(z\) of \(x\) and \(y.\)
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