Question

If \[a = c\] show that \[cba - abc = 0\]

Answer 1

Hint: Here we have three variables a, b and c. we have given that \[a = c\]. Therefore,
\[ \Rightarrow cba = bca = b \times c \times a = b \times a \times a = b{a^2}\]
\[ \Rightarrow abc = acb = a \times c \times b = a \times a \times b = {a^2}b\]

Complete step-by-step answer:
We have given \[a = c\]
 \[cba - abc = 0\]
Taking the given value by applying operation and properties we will proceed to the solution.
\[ \Rightarrow a = c\]
Multiply b to both sides, we get
\[ \Rightarrow a \times b = c \times b\]
 Multiply c to both sides, we get
$ \Rightarrow a \times b \times c = c \times b \times c$
As \[a = c\], we can replace c with a from R.H.S, we get
$ \Rightarrow a \times b \times c = c \times b \times a$
$ \Rightarrow abc = cba$ or
$ \Rightarrow cba = abc$
Subtract abc from both sides, we get
\[ \Rightarrow cba - abc = 0\]
 Hence, Proved.

Additional Information: - Properties of multiplication are as follows:
Closure property: - If we have given a and b as integers, then $a \times b$ is also an integer.
Commutative property: - if we have given a and b as integers, then $a \times b$
$ \Rightarrow a \times b = b \times a$
Associative property: - if we have given a, b and c as integers, then
$ \Rightarrow (a \times b) \times c = a \times (b \times c)$
Identity property: - if we have given a as integers, then
$ \Rightarrow a \times 1 = 1 \times a = a$
Zero property: - if we have given a as integers, then
$ \Rightarrow a \times 0 = 0 \times a = 0$
Distributive property:- if we have given a, b and c as integers, then
$ \Rightarrow a \times (b + c) = (a \times b) + (a \times c)$


Note: Alternate method: -
\[ \Rightarrow cba - abc = 0\]
We can write cba as abc i.e.
$
   \Rightarrow cba = c \times (b \times a) = c \times (a \times b) = (c \times a) \times b \\
   \Rightarrow abc = (a \times b) \times c = c \times (a \times b) = (c \times a) \times b \\
$ ………… by commutative property
As \[a = c\], therefore
$
   \Rightarrow (c \times a) \times b = (c \times c) \times b = {c^2}b \\
   \Rightarrow cba - abc = {c^2}b - {c^2}b = 0 \\
$