Convert to Parametric equation

[EV33]

Homework Statement

Are these systems of linear equations coincident planes, two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line.
X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

Homework Equations

X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

The Attempt at a Solution

X₃=X₁+2 X₂-2=3-X₁-X₂, I set them both equal to X₃and then each other

X₂=(5-2X₁)/3 , I am assuming this is the line of intersection.

My problem now is that I don't know how to convert to a parametric equation.
Could anyone please explain to me how to make a parametric equation.

 

[Altabeh]

Homework Statement

Are these systems of linear equations coincident planes, two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line.
X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

Homework Equations

X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

The Attempt at a Solution

X₃=X₁+2 X₂-2=3-X₁-X₂, I set them both equal to X₃and then each other

X₂=(5-2X₁)/3 , I am assuming this is the line of intersection.

My problem now is that I don't know how to convert to a parametric equation.
Could anyone please explain to me how to make a parametric equation.

0- Go through the test of finding the stance of two planes relative to each other.
1- If not parallel or coincident, find the normal vectors of both planes.
2- Compute their cross product. The resulting vector is parallel to the directorial vector of the intersecting line.
3- Here you just need to find a point from that line. (Find a joint point of both planes.)
4- Use the general equation of a line in 3d space and you are done.

AB

 

[HallsofIvy]

Homework Statement

Are these systems of linear equations coincident planes, two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line.
X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

Homework Equations

X₁+2 X₂-X₃=2
X₁+X₂+X₃=3

The Attempt at a Solution

X₃=X₁+2 X₂-2=3-X₁-X₂, I set them both equal to X₃and then each other

X₂=(5-2X₁)/3 , I am assuming this is the line of intersection.

My problem now is that I don't know how to convert to a parametric equation.
Could anyone please explain to me how to make a parametric equation.

First, do you understand that there exist an infinite number of parametric equations describing the same figure? The is not one correct set of parametric equations.

If, as in this case, you can set two of the variables equal to functions of the third, you can always use that third variable as the parameter. Here you have already solved for X₂ as a function of X₁ so use X₁ as the parameter- if you like you can write "X₁= t" and then "X₂= (5- 2t)/3. Put both of those into either X₁+2 X₂-X₃=2 or X₁+X₂+X₃=3 and solve for X₃ as a function of t. Those will be your parametric equations.