Critical Algebraic Thinking

Algebra Level 3

$\large {\begin{cases} x^y=1 \\ x+y=7 \end{cases}}$

What are the solutions to the system of equations above? Suppose the sum of all solutions for $x$ is $A$ , and the sum of all solutions for $y$ is $B$ . What is $A \times B$ ?

The answer is 98.

1 solution

Jesse Li
Feb 10, 2019

There are 3 ways a number taken to a power can equal 1:

The base is 1
The exponent is 0 and the base isn't 0
The base is -1 and the exponent is even

Therefore, the solutions are:

$x=1$ , $y=6$
$x=7$ , $y=0$
$x=-1$ , $y=8$

So, $A=7$ and $B=14$ . $7 \times 14= \boxed {98}$

I overlooked the negative solution :(

Peter van der Linden - 2 years, 4 months ago

Yeah, many people forget that possibility. It's important to remember that part when solving problems involving a number taken to a power being equal to 1.

Jesse Li - 2 years, 4 months ago

I too didn't consider that x = -1

Ram Mohith - 2 years, 4 months ago

@Ram Mohith – same that's why I got 48 lol

Charley Shi - 2 years, 3 months ago

Critical Algebraic Thinking

The answer is 98.

1 solution

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