System of Two Linear Equations with Two Variables. Second-Order Determinants

In theory of systems of linear equations it is convenient to use notion of determinant.

Second-order determinant is a number, that is defined by equality `|[a_(11),a_(12)],[a_(21),a_(22)]|=a_(11)a_(22)-a_(21)a_(12)`.

Numbers `a_(11),a_(22),a_(21),a_(22)` are called elements of the determinant; `a_(11)` and `a_(22)` are on the main diagonal, `a_(12)` and `a_(21)` are off the main diagonal. Therefore, second-order determinant equals product of elments on the main diagonal minus product of elements off the main diagonal.

For example, `|[-2,7],[-3,8]|=(-2)*8-(-3)*7=-16+21=5`.

Suppose we are given system of two linear equations with two variables: `{(a_(11)x+a_(12)y=b_1),(a_(21)x+a_(22)y=b_2):}`.

  1. If determinant of the system `Delta=|[a_(11),a_(12)],[a_(21),a_(22)]|!=0`, then system has unique solution, which can be found using Cramer's formula:

    `x=(|[b_1,a_(12)],[b_2,a_(22)]|)/(Delta)=(Delta_x)/(Delta)`, `y=(|[a_(11),b_1],[a_(21),b_2]|)/(Delta)=(Delta_y)/(Delta)`.

    Here determinant `Delta_x` is obtained from the determinant of the system by replacing first column with column of free members. Determinant `Delta_y` is obtained from the determinant of the system by replacing second column with column of free members.

  2. If `Delta=0`, but `Delta_x!=0` (or `Delta_y !=0`) then system doesn't have solutions (system is inconsistent).
  3. If `Delta=Delta_x=Delta_y=0` then system has infinitely many solutions.

Example 1 . Solve system `{(5x-2y=7),(10x+7y=3):}`.

Here `Delta=|[5,-2],[10,7]|=5*7-(-2)*10=35+20=55`.

`Delta_x=|[7,-2],[3,7]|=7*7-(-2)*3=49+6=55`.

`Delta_y=|[5,7],[10,3]|=5*3-7*10=15-70=-55`.

Therefore, `x=(Delta_x)/(Delta)=(55)/55=1` and `y=(Delta_y)/(Delta)=(-55)/(55)=-1`.

Thus, (1;-1) is solution of the system. Geometrically this means that lines, that are defined by equations of the system, intersect at point (1;-1).

Example 2 . Solve system `{(5x-2y=4),(35x-14y=200):}`.

Here `Delta=|[5,-2],[35,-14]|=5*(-14)-(-2)*35=-70+70=0`.

`Delta_x=|[4,-2],[200,-14]|=4*(-14)-(-2)*200=-56+400=344!=0`.

Since `Delta =0` and `Delta_x!=0` then system is inconsistent, i.e. it doesn't have solutions.

We could do this conclusion without determinants. Note that coeffcients near variables are proportional, but free members are not proportional: `5/35=(-2)/(-14)!=4/200`.

Geometrically this means that lines, that are defined by equations of the system are parallel.

Example 3. Solve system `{(0.2x+3.1y=-2.3),(x+15.5y=-11.5):}`.

Here `Delta=|[0.2,3.1],[1,15.5]|=0.2*15.5-3.1*1=3.1-3.1=0`.

`Delta_x=|[-2.3,3.1],[-11.5,15.5]|=(-2.3)*15.5-3.1*(11.5)=-35.65+35.65=0`.

`Delta_y=|[0.2,-2.3],[1,-11.5]|=0.2*(-11.5)-(-2.3)*1=-2.3+2.3=0`.

Since all determinants equal zero, system has infinitely many solutions.

We could do this conclusion without determinants. Note that coeffcients near variables and free members are proportional: `(0.2)/(1)=(3.1)/(15.5)=(-2.3)/(-11.5)`

Geometrically this means that lines, that are defined by equations of the system are same (match).