Is Ax = B consistent or inconsistent ?

Answer 1

The system is consistent. See explanation.

The system of equations #Ax=B# is consistent if #detA!=0#. Here we have:
#detA=|(1,0,6),(-2,8,5),(6,5,1)|=#
#=1*8*1+0*5*1+6*(-2)*(-2)-6*8*1-5*(-2)*1-1*0*(-2)=#
#8+24-48+10+0=-6#
#detA!=0#, so the system is consistent.
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Answer 2

The system is consistent.

Here's an approach without explicitly using determinants.

The augmented matrix #(A|b)# is given by #((1,0,6,|,1),(-2,8,5,|,3),(1,-2,1,|,-1))#
#R_2 -> R_2+2R_1, R_3 -> R_3-R_1 #: #((1,0,6,|,1),(0,8,17,|,5),(0,-2,-5,|,-2))#
#R_2 -> R_2/8# #((1,0,6,|,1),(0,1,17/8,|,5/8),(0,-2,-5,|,-2))#
#R_3 -> R_3+2R_2# #((1,0,6,|,1),(0,1,17/8,|,5/8),(0,0,-3/4,|,-3/4))#
#R_3 -> R_3/(-3/4)# #((1,0,6,|,1),(0,1,17/8,|,5/8),(0,0,1,|,1))#
Both the augmented matrix #(A|b)# and the coefficient matrix #A# have a rank of 3 - so the system is consistent.
As an added advantage, this method gives a direct way of finding the solution as well. #x_3 = 1#, # x_2+17/8x_3=5/8 implies x_2 = -3/2# #x_1+6x_3 = 1 implies x_1 = -5#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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