1. Vectors - Parametric Equations

Earth is described by the equation : x + y + z = 18
Mars is described by the equation: 4x + 3y - z = -3

Earth and Mars intersect each other

a/ find a parametric equation of the line in which earth intersect mars

b/ The coordinate of Spring Lake is given by the point (-5, 10, 13)
How far away from spring lake from the line mentioned in (a)?

2. Hello, shiiganB!

Astronomically, the problem makes no sense,
. . but I'll solve it anyway.

$\begin{array}{ccccc}\text{Earth is described by the equation:} & x + y + z &=& 18 & [1] \\
\text{Mars is described by the equation:} & 4x + 3y - z &=& \text{-}3& [2] \end{array}$

Earth and Mars intersect each other.

(a) Find a parametric equation of the line in which Earth intersect Mars.

Add [1] and [2]: . $5x + 4y \:=\:15 \quad\Rightarrow\quad x \:=\:3-\tfrac{4}{5}y$

Substitute into [1]: . $\left(3-\tfrac{4}{5}y\right) + y + z \:=\:18 \quad\Rightarrow\quad z \:=\:15-\tfrac{1}{5}y$

. . We have: . $\begin{Bmatrix}x &=& 3-\frac{4}{5}y \\ y &=& y \\ z &=& 15-\frac{1}{5}y \end{Bmatrix}$

On the right, replace $y$ with the parameter $t.$

. . Therefore: . $\begin{Bmatrix}x &=& 3 - \frac{4}{5}t \\ y &=& t \\ z &=& 15 - \frac{1}{5}t \end{Bmatrix}$

(b) The coordinates of Spring Lake is given by the point (-5, 10, 13).
How far away from Spring Lake is the line mentioned in (a)?
A strange (silly!) question . . .

Spring Lake could be a ski resort located on Mars or Jupiter or . . .

Ignoring that possibility, Spring Lake is probably located on Earth,
. . in which case, it is on the line mentioned in (a).

. . The distance is zero.

To check, try $t = 10.$