Think!!

Algebra Level 2

If $abc=xyz$ and $b=y$ , $c=z$ , then it is certain that $a=x$ .

False True

3 solutions

Vinayak Srivastava
May 11, 2020

There are four possibilities when $a\neq{x}$ :

$Case 1: b=y=0$ , then any values of $a, x$ will give $abc=xyz$ .

$Case 2: c=z=0$ , then any values of $a, x$ will give $abc=xyz$ .

$Case 3: a=0$ and any or both of $y,z=0$ , then any values of $0, x$ will give $abc=xyz$ .

$Case 4: x=0$ and any or both of $b,c=0$ , then any values of $a, 0$ will give $abc=xyz$ .

Hence, $a=x$ is not always true.

The explanation is alright. However, the question asked is incomplete/misleading: True or False? If abc=xyz and b=y, c=z, then a=x

Nowhere in the question it uses the term: always true or always false. So if I go by the question text, then a=x true is correct for many cases when 0 is not involved. so how is false an answer ? (unless you said always true).

Awadhesh Pratap - 1 year, 1 month ago

A Former Brilliant Member
May 11, 2020

$ayz=xyz\implies (a-x) yz=0$ . Hence there are eight possibilities :

(1) $a\neq 0,b=y=0,c=z\neq 0,x\neq 0,a\neq x$

(2) $a\neq 0,b=y\neq 0,c=z=0,x\neq 0,a\neq x$

(3) $a-x=0\implies a=x,b=y=0,c=z=0$

(4) $a-x=0\implies a=x,b=y=0,c=z\neq 0$

(5) $a-x=0\implies a=x,b=y\neq 0,c=z=0$

(6) $a-x=0\implies a=x,b=y\neq 0,c=z\neq 0$

(7) $a=0,b=y=0, c=z\neq 0,x\neq 0$

(8) $a=0,b=y=0,c=z=0,x\neq 0$

So the given equation doesn't necessarily imply that $a$ is equal to $x$

Cantdo Math
May 12, 2020

$b,y,c,z$ all are $0$ .Then the conclusion is false

Think!!

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