Question

Find a parametric representation of the solution set of the linear equation. (Enter your answer as a comma-separated list of equations. Use s and t as your parameters.) −8x + 5y − 4z = 1

Answer 1

`\left\{\begin{array}{ccc}x&=&(5)/(8)t-(1)/(2)s-(1)/(8)\\y&= & t\\z&= & s\end{array}`

Explanation:

There are at least 3 ways to do this problem, let us do it one way and then I'll explain how to do it from another approach. First of all let's begin by solving the linear equation for x. We get the following:

`x=(5)/(8)t-(1)/(2)s-(1)/(8)`

We can see that
`y` and
`z` could act as free variables as we have no conditions or restrictions on the values of x and y. Normally one would see such restrictions by adding more equations as in a system of equations. That is why we can call
`y` and
`z` free variables.

If we for instance let
`x=y` and
`z=s` where
`t,s\in \R` then we can write the linear equation as follows:

`\left\{\begin{array}{ccc}x&=&(5)/(8)t-(1)/(2)s-(1)/(8)\\y&= & t\\z&= & s\end{array}`

`t` and
`s` are calle parametric variables.