Plotting Symbolic 3x3 System of Equations

I have solved the following 3x3 system of equation

 %Define symbolic variables and equations
syms x y w
eq1 = 2*x -y + w == -5;
eq2 = y + 3*w == 7;
eq3 = w==2;

% 2. Solve the system
[xsol, ysol, wsol] = solve([eq1, eq2, eq3], [x, y,w])

xsol = 

ysol = 

1

wsol = 

2

I would like to produce the 3D solution for a Linear Algebra textbook. Is there any more effective way than the following? This textbook is for high school students

% Numeric intersection point
pt = double([S.x, S.y, S.w]);

Unable to resolve the name 'S.x'.

% Define plotting grid
[xg, yg] = meshgrid(linspace(pt(1)-5, pt(1)+5, 30), linspace(pt(2)-5, pt(2)+5, 30));

% Plane 1: 2*x - y + z = -5  -> z = -5 - 2*x + y
z1 = -5 - 2*xg + yg;

% Plane 2: y + 3*z = 7 -> z = (7 - y)/3
z2 = (7 - yg) / 3;

% Plane 3: z = 2
z3 = 2*ones(size(xg));

% Plot
figure
hold on
surf(xg, yg, z1, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.8 0.4 0.4])
surf(xg, yg, z2, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.4 0.8 0.4])
surf(xg, yg, z3, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.4 0.4 0.8])

xlabel('x'), ylabel('y'), zlabel('w')
legend('Plane 1','Plane 2','Plane 3','Location','best')
axis equal
grid on
view(3)
hold off

[Walter Roberson]

 %Define symbolic variables and equations
syms x y w
eq1 = 2*x -y + w == -5;
eq2 = y + 3*w == 7;
eq3 = w==2;

% 2. Solve the system
S = solve([eq1, eq2, eq3], [x, y,w])

S = struct with fields:
    x: -3
    y: 1
    w: 2

% Numeric intersection point
pt = double([S.x, S.y, S.w]);

% Define plotting grid
[xg, yg] = meshgrid(linspace(pt(1)-5, pt(1)+5, 30), linspace(pt(2)-5, pt(2)+5, 30));

% Plane 1: 2*x - y + z = -5  -> z = -5 - 2*x + y
z1 = -5 - 2*xg + yg;

% Plane 2: y + 3*z = 7 -> z = (7 - y)/3
z2 = (7 - yg) / 3;

% Plane 3: z = 2
z3 = 2*ones(size(xg));

% Plot
figure
hold on
surf(xg, yg, z1, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.8 0.4 0.4])
surf(xg, yg, z2, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.4 0.8 0.4])
surf(xg, yg, z3, 'FaceAlpha',0.5, 'EdgeColor','none', 'FaceColor',[0.4 0.4 0.8])

xlabel('x'), ylabel('y'), zlabel('w')
legend('Plane 1','Plane 2','Plane 3','Location','best')
axis equal
grid on
view(3)
hold off

Answer 1

[Steven Lord]

 %Define symbolic variables and equations
syms x y w
eq1 = 2*x -y + w == -5;
eq2 = y + 3*w == 7;
eq3 = w==2;
equations = [eq1, eq2, eq3];
fimplicit3(equations)

Now this doesn't look like the plot you showed. There's a reason for that; the ordering of the variables is different than you expect.

s = symvar(equations);
xlabel(string(s(1)))
ylabel(string(s(2)))
zlabel(string(s(3)))

If you need w to be along the "z" axis you could change the view on the axes.

Answer 2

[Sam Chak]

Hi @Athanasios Paraskevopoulos

This is a meaningful contribution toward writing a linear algebra textbook for high school students. When three non-parallel planes intersect, there are three possible scenarios. You have demonstrated the case of one unique solution (a single intersection point). Here are the other two scenarios: infinite solutions and no solution.

Note that I can't use fsurf in these two scenarios because MATLAB assumes that any expression corresponds to z. Therefore, I use fimplicit3 instead.

syms x y z

% ===============================
%% Scenario 2: Infinite Solutions
% ===============================
% 3D plotting limits [xmin, xmax, ymin, ymax, zmin, zmax] for Scenario 2
limit2 = [0, 5.01, -4.01, 4.01, -5, 5];

figure
hold on
fimplicit3(x - y    == 0, limit2, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.8 0.4 0.4]);
fimplicit3(x + y    == 0, limit2, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.4 0.8 0.4]);
fimplicit3(x - 10*y == 0, limit2, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.4 0.4 0.8]);
hold off
grid on

view(3)
xlabel('X Axis')
ylabel('Y Axis')
zlabel('Z Axis')
title('The Binding of a Book (Infinite Solutions)')

% ===============================
%% Scenario 3: No Solution
% ===============================
% 3D plotting limits for Scenario 3
limit3 = [1, 5.01, -4.01, 4.01, -5, 5];

figure
hold on
fimplicit3(x + y == 1, limit3, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.8 0.4 0.4]);
fimplicit3(x - y == 1, limit3, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.4 0.8 0.4]);
fimplicit3(x     == 5, limit3, 'FaceAlpha', 0.25, 'EdgeColor', 'none', 'FaceColor', [0.4 0.4 0.8]);
hold off
grid on

view(3)
xlabel('X Axis')
ylabel('Y Axis')
zlabel('Z Axis')
title('A Hollow Triangular Prism (No Solution)')