# If x+y=19 and xy=78, find y-x?

##### 1 Answer
Jan 30, 2017

$y - x = \pm 7$

#### Explanation:

As $x + y = 19$ and $x y = 78$, we have

${\left(x + y\right)}^{2} = {x}^{2} + 2 x y + {y}^{2}$

i.e. ${19}^{2} = {x}^{2} + {y}^{2} + 2 \times 78$

i.e. ${x}^{2} + {y}^{2} = {19}^{2} - 2 \times 78 = 361 - 156 = 205$

Hence, ${\left(y - x\right)}^{2} = {y}^{2} + {x}^{2} - 2 x y = 205 - 2 \times 78 = 205 - 156 = 49$

and $y - x = \pm 7$

In fact we can solve for $x$ and $y$ and the two numbers will be $13$ and $6$ or $6$ and$13$.