Two equations

Algebra Level 2

If both $a + b = c$ and $a^{2} + b^{2} = c^{2}$ are true and $a < b$ , what is the value of $a + (b - c)$ ?

The answer is 0.

2 solutions

Vilakshan Gupta
Mar 16, 2018

Squaring $a+b=c$ $\implies$ $a^2+2ab+b^2=c^2 \implies c^2+2ab=c^2 \implies 2ab=0$

Either $a$ or $b$ must be $0$ , but since $a<b$ , we can have $a=0$ .

Then if $a=0$ , this implies $b=c$ .

Then $b-c=0$ and hence $a+(b-c)=\boxed{0}$

If $b=0$ and $a<0$ , then we have $a=c$ and hence $a+(b-c)=a-c=\boxed{0}$

Nice solution. Since we are not given conditions on $a,b$ we could also have $b = 0$ , (making $a \lt 0$ ), but this would just lead to $a = c$ and $a + (b - c) = a - c = 0$ as well.

Brian Charlesworth - 3 years, 2 months ago

Thanks. I have edited the solution.

Vilakshan Gupta - 3 years, 2 months ago

Ivan Jambrešić
Oct 18, 2018

$a+(b-c)=a+b-c$

$c=a+b$

$a+b-c=a+b-(a+b)=a+b-a-b=0$

Two equations

The answer is 0.

2 solutions

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