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Matemática discreta Exemplos
5x-5y+2z=-135x−5y+2z=−13 , 10x-10y+4z=-2310x−10y+4z=−23 , 15x-15y+6z=-3715x−15y+6z=−37
Etapa 1
Write the system as a matrix.
[5-52-1310-104-2315-156-37]⎡⎢
⎢⎣5−52−1310−104−2315−156−37⎤⎥
⎥⎦
Etapa 2
Etapa 2.1
Multiply each element of R1R1 by 1515 to make the entry at 1,11,1 a 11.
Etapa 2.1.1
Multiply each element of R1R1 by 1515 to make the entry at 1,11,1 a 11.
[55-5525-13510-104-2315-156-37]⎡⎢
⎢
⎢⎣55−5525−13510−104−2315−156−37⎤⎥
⎥
⎥⎦
Etapa 2.1.2
Simplifique R1R1.
[1-125-13510-104-2315-156-37]⎡⎢
⎢⎣1−125−13510−104−2315−156−37⎤⎥
⎥⎦
[1-125-13510-104-2315-156-37]⎡⎢
⎢⎣1−125−13510−104−2315−156−37⎤⎥
⎥⎦
Etapa 2.2
Perform the row operation R2=R2-10R1 to make the entry at 2,1 a 0.
Etapa 2.2.1
Perform the row operation R2=R2-10R1 to make the entry at 2,1 a 0.
[1-125-13510-10⋅1-10-10⋅-14-10(25)-23-10(-135)15-156-37]
Etapa 2.2.2
Simplifique R2.
[1-125-135000315-156-37]
[1-125-135000315-156-37]
Etapa 2.3
Perform the row operation R3=R3-15R1 to make the entry at 3,1 a 0.
Etapa 2.3.1
Perform the row operation R3=R3-15R1 to make the entry at 3,1 a 0.
[1-125-135000315-15⋅1-15-15⋅-16-15(25)-37-15(-135)]
Etapa 2.3.2
Simplifique R3.
[1-125-13500030002]
[1-125-13500030002]
Etapa 2.4
Multiply each element of R2 by 13 to make the entry at 2,4 a 1.
Etapa 2.4.1
Multiply each element of R2 by 13 to make the entry at 2,4 a 1.
[1-125-135030303330002]
Etapa 2.4.2
Simplifique R2.
[1-125-13500010002]
[1-125-13500010002]
Etapa 2.5
Perform the row operation R3=R3-2R2 to make the entry at 3,4 a 0.
Etapa 2.5.1
Perform the row operation R3=R3-2R2 to make the entry at 3,4 a 0.
[1-125-13500010-2⋅00-2⋅00-2⋅02-2⋅1]
Etapa 2.5.2
Simplifique R3.
[1-125-13500010000]
[1-125-13500010000]
Etapa 2.6
Perform the row operation R1=R1+135R2 to make the entry at 1,4 a 0.
Etapa 2.6.1
Perform the row operation R1=R1+135R2 to make the entry at 1,4 a 0.
[1+135⋅0-1+135⋅025+135⋅0-135+135⋅100010000]
Etapa 2.6.2
Simplifique R1.
[1-125000010000]
[1-125000010000]
[1-125000010000]
Etapa 3
Use the result matrix to declare the final solution to the system of equations.
x-y+25z=0
0=1
0=0
Etapa 4
The solution is the set of ordered pairs that make the system true.
(y-2z5,y,z)