# 2positive integers have a sum of 10 and a product of 21, what are the integers?

May 15, 2018

$x = 3$ or $x = 7$
$y = 7$ or $y = 3$

#### Explanation:

Lets say two positive integers are $x$ and $y$.

So $x + y = 10$ and $x \times y = 21$

We have two equations here:
$x + y = 10$ ----> equation 1
$x y = 21$ ----> equation 2

From equation 1, make $y$ the subject.

$x + y = 10$
$y = 10 - x$ --- substitute $y$ in equation 2

$x y = 21$ ---- Equation 2

x(10-x) = 21 10x-x^2 = 21#

Re-arranging the equation, we get:
$- {x}^{2} + 10 x - 21 = 0$

Lets solve for $x$ now:

$- {x}^{2} + 10 x - 21 = 0$

Multiply by ($- 1$) throughout and we get:

${x}^{2} - 10 x + 21 = 0$

Factors are $- 3$ and $- 7$ as $- 3 + - 7 = 10$ and $- 3 \times \left(- 7\right) = 21$

${x}^{2} - 10 x + 21 = 0$
${x}^{2} - 3 x - 7 x + 21 = 0$
$x \left(x - 3\right) - 7 \left(x - 3\right) = 0$
$\left(x - 7\right) \left(x - 3\right) = 0$

Hence, $x = 7$ or $x = 3$

Substitute value of $x$in equation 1

Take $x = 7$
$7 + y = 10$
$y = 10 - 7$
$y = 3$ -----> First value of $y$

Take $x = 3$

$3 + y = 10$
$y = 10 - 3$
$y = 7$ ------> Second value of $y$

$x = 3$ or $x = 7$
$y = 7$ or $y = 3$