# Linear Equations using matrix and variables on the line

Q. I need a find a system of linear equations with three unknown variables whose solutions are the points on the line through (1,1,1) and (3,5,0).

$\frac{x-1}{2} = \frac{y-1}{4} = \frac{z-1}{-1}$ and I set them equal to t that is an arbitrary constant.

Then $x = 2t + 1, y = 4t + 1 , z = -t + 1$ becomes the solutions of the linear equation systems?

Is it correct? From here I know how to do it. Just wanted to make sure the intro solution was correct. Please let me know! Thanks!

## 2 Answers

You are right about the parametric equation(s) of the line.

However, to answer the question as put, you need to eliminate $t$. So for example you could solve for $t$ in terms of $z$, and substitute this expression for $t$ in your first two equations. You will get a system of $2$ equations in the unknowns $x$, $y$, and $z$.

Remark: There are many systems of two equations that will do the job. Geometrically, they are the equations of any two distinct planes that contain the line.

Yes, that is a correct parametrization of the line.