Insira um problema...
Matemática discreta Exemplos
-10x-15y-z=14−10x−15y−z=14 , -6x-25y-2z=55−6x−25y−2z=55 , 8x+15y+z=-20
Etapa 1
Write the system as a matrix.
[-10-15-114-6-25-2558151-20]
Etapa 2
Etapa 2.1
Multiply each element of R1 by -110 to make the entry at 1,1 a 1.
Etapa 2.1.1
Multiply each element of R1 by -110 to make the entry at 1,1 a 1.
[-110⋅-10-110⋅-15-110⋅-1-110⋅14-6-25-2558151-20]
Etapa 2.1.2
Simplifique R1.
[132110-75-6-25-2558151-20]
[132110-75-6-25-2558151-20]
Etapa 2.2
Perform the row operation R2=R2+6R1 to make the entry at 2,1 a 0.
Etapa 2.2.1
Perform the row operation R2=R2+6R1 to make the entry at 2,1 a 0.
[132110-75-6+6⋅1-25+6(32)-2+6(110)55+6(-75)8151-20]
Etapa 2.2.2
Simplifique R2.
[132110-750-16-7523358151-20]
[132110-750-16-7523358151-20]
Etapa 2.3
Perform the row operation R3=R3-8R1 to make the entry at 3,1 a 0.
Etapa 2.3.1
Perform the row operation R3=R3-8R1 to make the entry at 3,1 a 0.
[132110-750-16-7523358-8⋅115-8(32)1-8(110)-20-8(-75)]
Etapa 2.3.2
Simplifique R3.
[132110-750-16-7523350315-445]
[132110-750-16-7523350315-445]
Etapa 2.4
Multiply each element of R2 by -116 to make the entry at 2,2 a 1.
Etapa 2.4.1
Multiply each element of R2 by -116 to make the entry at 2,2 a 1.
[132110-75-116⋅0-116⋅-16-116(-75)-116⋅23350315-445]
Etapa 2.4.2
Simplifique R2.
[132110-7501780-233800315-445]
[132110-7501780-233800315-445]
Etapa 2.5
Perform the row operation R3=R3-3R2 to make the entry at 3,2 a 0.
Etapa 2.5.1
Perform the row operation R3=R3-3R2 to make the entry at 3,2 a 0.
[132110-7501780-233800-3⋅03-3⋅115-3(780)-445-3(-23380)]
Etapa 2.5.2
Simplifique R3.
[132110-7501780-2338000-116-116]
[132110-7501780-2338000-116-116]
Etapa 2.6
Multiply each element of R3 by -16 to make the entry at 3,3 a 1.
Etapa 2.6.1
Multiply each element of R3 by -16 to make the entry at 3,3 a 1.
[132110-7501780-23380-16⋅0-16⋅0-16(-116)-16(-116)]
Etapa 2.6.2
Simplifique R3.
[132110-7501780-233800011]
[132110-7501780-233800011]
Etapa 2.7
Perform the row operation R2=R2-780R3 to make the entry at 2,3 a 0.
Etapa 2.7.1
Perform the row operation R2=R2-780R3 to make the entry at 2,3 a 0.
[132110-750-780⋅01-780⋅0780-780⋅1-23380-780⋅10011]
Etapa 2.7.2
Simplifique R2.
[132110-75010-30011]
[132110-75010-30011]
Etapa 2.8
Perform the row operation R1=R1-110R3 to make the entry at 1,3 a 0.
Etapa 2.8.1
Perform the row operation R1=R1-110R3 to make the entry at 1,3 a 0.
[1-110⋅032-110⋅0110-110⋅1-75-110⋅1010-30011]
Etapa 2.8.2
Simplifique R1.
[1320-32010-30011]
[1320-32010-30011]
Etapa 2.9
Perform the row operation R1=R1-32R2 to make the entry at 1,2 a 0.
Etapa 2.9.1
Perform the row operation R1=R1-32R2 to make the entry at 1,2 a 0.
[1-32⋅032-32⋅10-32⋅0-32-32⋅-3010-30011]
Etapa 2.9.2
Simplifique R1.
[1003010-30011]
[1003010-30011]
[1003010-30011]
Etapa 3
Use the result matrix to declare the final solution to the system of equations.
x=3
y=-3
z=1
Etapa 4
The solution is the set of ordered pairs that make the system true.
(3,-3,1)