0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
6
  • $\begingroup$ 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. $\endgroup$ Jan 10 '18 at 4:55
  • $\begingroup$ 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 $\endgroup$ Jan 10 '18 at 4:57
  • $\begingroup$ Sorry I misunderstood your question $\endgroup$ Jan 10 '18 at 4:57
  • $\begingroup$ 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? $\endgroup$ Jan 10 '18 at 5:07
  • $\begingroup$ This is the standard equation of a line between two points. Convince yourself it is straight and that it passes through the points $\endgroup$ Jan 10 '18 at 5:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.