# 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).