Simultaneous Equations

- Linear Algebra 

Consider the following set of equations for our example.

-6x = 2y - 2z + 15
 4y - 3z = 3x + 13
 2x + 4y - 7z = -9

First, rearrange the equations.

Write each equation with all the unknown variables on the left-hand side and all known quantities on the right side. Thus, for the equations given, rearrange them such that all terms involving x, y and z are on the left side of the equal sign.

-6x - 2y + 2z = 15
-3x + 4y - 3z = 13
 2x + 4y - 7z = -9

Second, write the equations in a matrix form

To write the equations in the matrix form Ax = b, where x is the vector of unknowns (you have to arrange the unknowns in vector x), the coefficients of the unknowns in matrix A and the constants on the rigth hand of the equations in vector b. In this particualar example, the unknown column  vector is

x =  [x y z]'

the coefficient matrix is

A = [-6 -2  2
     -3  4 -3
      2  4 -7]

and the known constant column vector is

b = [15 13 -9]'

Note than the columns of A simply are the coefficients of each unknown from all the three expressed equations. The apostrophe at the end of vectors x and b means that those vectors are column vectors, not row ones (it is Matlab notation).

Third, solve the simultaneous equations in Matlab

Enter the matrix A and vector b, and solve for vector x with the command 

x = A\b

(note that the '\' symbol is different from the ordinary division '/' sign).

The Matlab answer to the lines above is:

A =
    -6    -2     2
    -3     4    -3
     2     4    -7

b =
    15
    13
    -9

x =
   -2.7273
    2.7727
    2.0909

You can test the result by performing the substitution and multiplying Ax to get b, like this:

A*x

And the Matlab answer is:

ans =
   15.0000
   13.0000
   -9.0000

which corresponds to the column vector b, indeed.

Video Summary -  Solve Linear Systems