# How to use Parametric Equations to define the solution of a Linear System in 3 Variables

I am given the following Linear System of Equations:

$$x - y - z = 1$$ $$2x - z = 3$$ $$x - 7y - 4z = -2$$

I am told to show that $(1, 1, -1)$ and $(2, 0, 1)$ are valid solutions to this system which is easily done.

However, I am then asked to find the line that passes through those points and to give the equation of that line using Parametric Equations.

So I understand Parametric Equations, and that the (infinite) solutions of this system is a line where the the planes intersect, and that I'm being asked to convert the Linear System into related Parametric Equations that defines this solution-line. I cannot however, seem to figure out exactly what that needs to look like or how to get there.

The line between the points $$(1,1,-1)$$ and $$(2,0,1)$$ is given by $$r(t)=t(1,1,-1)+(1-t)(2,0,1)=(2-t,t,1-2t)$$ for $0\leq t\leq 1$.

If your question is how to define the line of solutions to a system of $n$ equations in $n$ unknowns with one degree of freedom the standard procedure is to eliminate $n-1$ of the variables and use the final variable as your parameter.

• My question was what you descried. I'm just not sure where exactly I need to end up, and so am having trouble getting started. I tried a bunch of factoring and elimination, but am not sure what the final form is going to look like and so am sailing without a compass so to speak. Jan 10 '18 at 4:55
• Oh in this case the line is the one I gave but allowing t to live in R. If you know the solution space is a line and you have two points on it that's enough to figure out exactly what the line is Jan 10 '18 at 4:57
• Sorry I misunderstood your question Jan 10 '18 at 4:57
• Right, but I'm not sure how you derived that final set of parametric equations (2-t, t, 1-2t). How do you get from the original System of Equations and the Two Points to that? Jan 10 '18 at 5:07
• This is the standard equation of a line between two points. Convince yourself it is straight and that it passes through the points Jan 10 '18 at 5:08