Question

If x and y are positive integers and x+y+xy=54, then x+y equals what

Answer 1

Answer:

x+y = 14

Step-by-step explanation:

x+y+xy = 54

1+x+y+xy = 55

(1+x)(1+y) = 55

(1+x)(1+y) = 5*11

If x and y are integers,

1+x = 5    => x = 4

1+y = 11   =>  y = 10

or

1+x = 11   => x = 10

1+y = 5   =>  y = 4

In either case, x+y = 14

Answer 2

The solution to this system is $x = 4$ and $y = 10$. Then, $x+y = 14$.

Let be $x + y + x\cdot y = 54$, then we proceed to modify the expression by algebraic means:

1. $x + y + x\cdot y = 54$  Given
2. $x+y+x\cdot y + 1 = 55$  Compatibility with addition/Definition of addition
3. $(x+1)+(y+x\cdot y) = 55$  Commutative property/Associative property
4. $(x+1) + y\cdot (x + 1) = 55$  Distributive property/Commutative property
5. $(x+1)\cdot (y +1) = 55$  Distributive property

The only divisors of 55 is 5 and 11, respectively. Hence, we have the following  system of linear equations:

$x + 1 = 5$ (1)

$y + 1 = 11$ (2)

The solution to this system is $x = 4$ and $y = 10$. Then, $x+y = 14$.

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