When can I use a parameter in equation (of the a plane)

Question

In my book there is an example: Find vector and parametric equation of the plane $x-y+2z=5$

Now, the solution is: solving for $x$ in terms of $y$ and $z$ yields $x = 5+y-2z$ and then using parameters $t_1$ and $t_2$ yields $x=5+t_1-2t_2$, $y=t_1$ and $z=t_2$

Now my questions is, when can I use parameters "just like that". Why could I use t1 and t2 here and why not t3 for x? If I have had equation with variables: $a,b,c,d,e,f,g,h$ would I represent $b,c,d,e,f,g,h$ using parameters too?

Thanks

Answer 1

I ill assume that you are asking only about the linear equations. In parametrization, independent variables are represented as parameters and dependent variables by their dependency on these variables. This method is especially useful when you have to plot the points on this curve using a program.

Here x is not represented by a new parameter (t3) because x is dependent on y and z, and hence can be represented in terms of t1 and t2. If you take x and y as independent variables, then z becomes the dependent variable. How do we find the number of independent variables in a linear equation?

In a linear equation like:

ax+by+cz=0 (known a,b,c constants and x,y,z variables)

The number of independent variables is the dimension of the null space of matrix [a b c]. When we have a series of equations, the number of dependent variables would still be the null space of the constant matrix.

When the equation becomes non-homogenous i.e.

ax+by+cz=d

we can find you can use augmented matrix and row reduce it to find the independent variables. The talk by Prof. Gilbert Strang on Linear Algebra for Engineers was found very helpful in understanding the linear dependence concepts.