<code>
% Story problem: 2 lbs carrots and 3lbs apples cost $10.15. 1 lb carrots and 
% 7 lb apples cost $17.12. How much do carrots and apples cost per pound?

% eqns are
% 2c + 3a = 10.15
% 1c + 7a = 17.12

% To solve in Matlab, write it in matrix-vector form as an Ax=b problem.
% A holds the coefficients of the unknowns in the left-hand sideof the eqn
% x is the vector of unknowns c and a
% b is the vector of known right-hand-side values 10.15 and 17.12

A = [2 3 ; 1 7]

A =
   2   3
   1   7

b = [10.15 ; 17.12]

% Solve Ax=b for x using Matlab's "backslash" operator
x = A\b

x =
    1.7900
    2.1900

% Verify that the solution satisfies the original equations by using
% long-winded high-school arithmetic. First assign the components of
% x into variables c and a.

c = x(1)
c =
    1.7900

a = x(2)
a =
    2.1900

% verify that 2c + 3a = 10.15
2*c + 3*a

ans =
   10.1500

% verify that c + 7a = 17.12
c + 7*a 

ans =
   17.1200

% Now verify the solution using matrix multiplication
% ie verify that Ax = b

A

A =
     2     3
     1     7

x

x =
    1.7900
    2.1900

A*x

ans =
   10.1500
   17.1200


% New problem
% 1c + 2a + 3t = 18.84
% 2c + 1a      = 5.97
%      6a + 2t = 23.52

% Enter this into matlab as an Ax=b problem
A = [ 1 2 3 ; 2 1 0 ; 0 6 2]

A =
     1     2     3
     2     1     0
     0     6     2

b = [18.84 ; 5.97 ; 23.52]

b =
   18.8400
    5.9700
   23.5200

% Solve Ax=b using the backslash operator
x = A\b

x =
    1.6900
    2.5900
    3.9900

% So the answer is 
% carrots  cost $1.69/lb
% apples   cost $2.59/lb
% tomatoes cost $3.99/lb

% Wasn't that a total breeze, compared to solving three equations in three
% unknowns by hand? Hooray for numerical linear algebra!

% Verify first equation long-windedly
c = x(1)

c =
    1.6900

a = x(2)

a =
    2.5900

t = x(3)

t =
    3.9900

1*c + 2*a + 3*t 

ans =
   18.8400

% Now verify all equations by multiplying A*x and comparingto b
A*x

ans =
   18.8400
    5.9700
   23.5200

A

A =
     1     2     3
     2     1     0
     0     6     2

x

x =
    1.6900
    2.5900
    3.9900

A*x

ans =
   18.8400
    5.9700
   23.5200

b

b =
   18.8400
    5.9700
   23.5200


% Observe above that A*x equals b. Can also verify Ax=b by computing
% Ax-b and seeing that it's zero
% that

A*x-b

ans =

   1.0e-14 *

    0.3553
         0
         0

% Note that Ax-b is not exactly zero. It has magnitude 1e-14, so it's 
% very nearly zero. That's because the numerical computations are done
% in finite precision (there's rounding error).

% Question: how do you do subscripts in Matlab?

A

A =

     1     2     3
     2     1     0
     0     6     2

% access the i,j = 3,2 element of A via A(3,2)
A(3,2)

ans =
     6

% Get second column of A
A(:,2) 

ans =

     2
     1
     6

% In matlab, A(:,j) means all rows in jth col
% In matlab, A(i,:) means all cols in ith row
A

A =
     1     2     3
     2     1     0
     0     6     2

A(1,:)

ans =
     1     2     3

A(2,:)

ans =
     2     1     0

A(3,:)

ans =
     0     6     2

A

A =
     1     2     3
     2     1     0
     0     6     2


% Change the 3,1 elem to 99
A(3,1) = 99

A =
     1     2     3
     2     1     0
    99     6     2

% Change the 3rd row to 7, 2, 19
A(3,:) = [7 2 19]


A =
     1     2     3
     2     1     0
     7     2    19

% Change the 2nd column to -1 4 6 
A(:,2) = [-1 ; 4 ; 6]

A =
     1    -1     3
     2     4     0
     7     6    19

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