Comments

Stepped Systems


We use Cramer's rule to discuss and solve linear systems in which the number of equations (m) is equal to the number of unknowns (no). When m and no are larger than three, it is very laborious to use this rule. Therefore, we use the technique of staggering, which facilitates the discussion and resolution of any linear systems.

We say that a system, where there is at least one nonzero coefficient in each equation, is staggered if the number of null coefficients before the first nonzero coefficient increases from equation to equation. To scale a system we adopted the following procedure:

a) We fix as 1st equation one of those that have the coefficient of the first unknown nonzero.

b) Using the properties of equivalent systems, we nullify all coefficients of the first unknown of the other equations.

c) We repeat the process with the other unknowns until the system becomes staggered.

Let's apply the scheduling technique, considering two types of system:

I. The number of equations equals the number of unknowns (m = n)

Example 1:

1st step: We nullify all coefficients of the 1st unknown from the 2nd equation by applying the properties of the equivalent systems:

  • We swap the 1st equation with the 2nd equation, so that the 1st coefficient of x equals 1.
  • We exchange the 2nd equation for the sum of the 1st equation, multiplied by -2, with the 2nd equation:

  • We exchange the 3rd equation for the sum of the 1st equation, multiplied by -3, with the 3rd equation:

2nd step: We nullify the coefficients of the 2nd unknown from the 3rd equation:

  • We exchange the 3rd equation for the sum of the 2nd equation, multiplied by -1, with the 3rd equation:

Now the system is staggered and we can solve it.

-2z = -6 z = 3

Replacing z = 3 in (II):

-7y - 3 (3) = -2 -7y - 9 = -2 y = -1

Replacing z = 3 and y = -1 in (I):

x + 2 (-1) + 3 = 3 x = 2

So x = 2, y = -1 and z = 3

Example 2:

1st step: We nullify all coefficients of the 1st unknown from the 2nd equation:

  • We exchange the 2nd equation for the sum of the product of the 1st equation for -2 with the 2nd equation:

  • We exchange the 3rd equation for the sum of the product of the 1st equation for -3 with the 3rd equation:

2nd step: We nullify the coefficients of the 2nd unknown from the 3rd equation:

  • We exchange the 3rd equation for the sum of the product of the 2nd equation for -1 with the 3rd equation:

That way the system is staggering. Since there is no real value of z such that 0z = -2, the system is impossible.

Next: Stepped Systems (continued)