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Graphically solve a system of equations and plot them in R^3?

Filip Fornell
Posted 5 years ago

Hello,

I have an assignment that asks me to graphically solve the system of equations i.e. plot them in ?^3 :

x + 2y + 3z = 1

2x + 4y + 7z = 2

3x + 7y + 11z = 8

Currently, I've used ContourPlot3D for plotting my planes in a 3-dimensional space and then FindRoot in order to find the solution to the system.

ContourPlot3D[{x + 2 y + 3 z == 1, 2 x + 4 y + 7 z == 2, 
  3 x + 7 y + 11 z == 8}, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}]
FindRoot[{x + 2 y + 3 z == 1, 2 x + 4 y + 7 z == 2, 
  3 x + 7 y + 11 z == 8}, {x, 0}, {y, 0}, {z, 0}]

My question here is if I've correctly completed the assignement and if I did, why would I need to plot the three planes in ?^3 when it doesn't really help me in solving the problem?
POSTED BY: Filip Fornell
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Filip Fornell
Filip Fornell
Posted 5 years ago

Is it mainly the creators of the assignement wanting us to familiarize ourselves with the different Mathematica functions?
POSTED BY: Filip Fornell

Is it mainly the creators of the assignement wanting us to familiarize ourselves with the different Mathematica functions?

No - simply by plotting you can see the solution, namely where all three planes intersect. You can set the focus on that spot:

sol = FindRoot[{x + 2 y + 3 z == 1, 2 x + 4 y + 7 z == 2, 3 x + 7 y + 11 z == 8}, {x, 0}, {y, 0}, {z, 0}];
reg = Sequence @@ ({#, # - 1 /. sol, # + 1 /. sol} & /@ {x, y, z});
ContourPlot3D[{x + 2 y + 3 z == 1, 2 x + 4 y + 7 z == 2, 3 x + 7 y + 11 z == 8}, Evaluate@reg]

enter image description here
POSTED BY: Henrik Schachner
Filip Fornell
Filip Fornell
Posted 5 years ago

I see. I'll remember that you can actually focus the view on the intersection point like you illustrated. Makes it way faster than rotating the graph and looking for an intersection.

Thank you for taking the time helping me.
POSTED BY: Filip Fornell
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