x1 - x2 - x3 = 7
x1 - x2 - 2x3 = 2
I know how to solve the problem, but I don't know how to show it geometrically so here is the answer:
= x2 +
How do you graph this? Help please
The graphs are planes in 3 space, and their intersection is a line.
To plot a plane, you need 3 points.
Easiest 3 points to find are on the axes (e.g. x1 when x2 = x3 =0, x2 when x1=x3=0, ...)
Connect the 3 points and you will have triangle in the plane you want to sketch.
Your solution can be given as a parameteric line (let x2 on the right hand side be t)
x1 = t + 12
x2 = t
x3 = 5
Since x3 is constant, the line is parallel to the x1-x2 plane.
You already have it in the answer. The starting point is (12,0,5) and the direction vector is (1,1,0)
but, as snowtea says, geometrically, a line equation in three variables corresponds to a plane so the solution set is the line of intersection of those planes.
One way to solve that is to set it up as an "augmented matrix" and row-reduce:How do you graph this? Help please
which says that , so that and . That line can be written as the parametric equations , , . Taking t= 0 gives (12, 0, 5) and taking t= 1 gives (13, 1, 5). The graph is the straight line through those two points.
Thank you Snowtea and Hallsofivy for responding
Again thanks for your help, but I still don't understand how to graph this:
Do you know how to plot points on a 3d cartesian (x,y,z) grid on a piece of paper?
Something like this:
Plot (12,0,5) and (13,1,5) in the 3d grid and connect them with a straight line. This will be the 2d projection of the line on your paper grid.
One more thing about a geometric solution.
The normal vectors for the two planes are:
(1, -1, -1)
(1, -1, -2)
just by looking at the coefficients.
The line at the intersection of both of them must be orthogonal to both these vectors.
So the vector parallel to the line should be a scalar multiple of the cross product of (1, -1, -1) and (1, -1, -2).
I think once you learn how to plot one point correctly you should be set.
So lets try (12,0,5).
Remember this is only a point, so the lines are only for reference only the final point location matters.
Start at (0,0,0)
Move parallel to x1 12 units (you did this correctly)
Move parallel to x2 0 units (stay where you are)
Move parallel to x3 5 units (directly move up 5 units)
You point should go here.
Look at the diagram I linked to before, and see if this makes sense.