Question

If we have two equations as (ab – b + 1=0) and (bc – c + 1 = 0) then what is (a – ac) equal to?
(A). -1
(B). 0
(C). 1
(D). 2

Answer 1

Hint: First solve the required term and separate the terms a, c and write it as a product of 2 terms. Now using the 2 equations you have, manipulate them and find the value of the product you got. Just solve algebraically all terms to get value from the simpler terms.

Complete step-by-step solution -
Required term for which, we need to find the value:
\[\Rightarrow a-ac\]
Now take a common from the term, we get the term:
\[\Rightarrow a\left( 1-c \right)\] ------- (1)
The first equation given in the question is written as:
\[\Rightarrow ab-b+1=0\] ------------ (2)
The second equation given in the question is written as:
\[\Rightarrow bc-c+1=0\] ----------- (3)
Let us assume the variable k ,whose value is equation (1):
\[\Rightarrow k=a\left( 1-c \right)\] ----- (4)
Now by subtracting 1 on both sides of equation (3):
\[\Rightarrow bc-c+1-1=-1\]
By simplifying, we get the above equation in the form:
\[\Rightarrow bc-c=-1\] ---------(5)
By subtracting term bc on both sides of equation (3):
\[\Rightarrow 1-c=-bc\] --------- (6)
By substituting equation (6) in equation (4), we get:
\[\Rightarrow k=a\left( -bc \right)=-abc\] ---- (7)
By subtracting 1 on both sides of equation (2), we get:
\[\Rightarrow ab-b=-1\]
By adding b on both sides of above equation, we get:
\[\Rightarrow a=\dfrac{b-1}{b}\] ---------(8)
By substituting equation (8) in equation (7), we get it as:
\[\Rightarrow k=-\dfrac{\left( b-1 \right)}{b}.bc\]
By simplifying the above equation, we get it in the form:
\[\Rightarrow k=-\left( b-1 \right)c\]
By simplifying more to above equation, we can write it in form of:
\[\Rightarrow k=-\left( bc-c \right)\]
By substituting equation (5), we get value of k as:
\[\Rightarrow k=-\left( -1 \right)=1\]
By substituting original value of k, we get the equation:
\[\Rightarrow a-ac=1\]
So, the required value of a given expression is 1.
Therefore, option (c) is the correct answer for a given question.

Note: These are the large number of equations. So, be careful with equation numbers, generally students forget the “-” sign into the term k. After substituting the value of “a” don’t forget the “-” sign because if you forget then you get the result as -1 which is also there in the question. So, be careful with this sign.