Question

Two positive integers have a sum of 10 and a product of 21. What are the integers?

Answer 1

3 and 7

Explanation:

`x,y-\text{positive integers}\\\\\text{The system of equations:}\\\\\left\{\begin{array}{ccc}x+y=10&\text{subtract}\ y\ \text{from both sides}\\xy=21\end{array}\right\\\\\left\{\begin{array}{ccc}x=10-y&(1)\\xy=21&(2)\end{array}\right\\\\\text{substitute (1) to (2):}\\\\(10-y)(y)=21\qquad\text{use the distributive property}\ a(b-c)=ab-ac\\\\10y-y^2=21\qquad\text{subtract 21 from both sides}\\\\-y^2+10y-21=0\qquad\text{change the signs}\\\\y^2-10y+21=0\\\\y^2-3y-7y+21=0\\\\y(y-3)-7(y-3)=0`

`(y-3)(y-7)=0\iff y-3=0\ \vee\ y-7=0\\\\y-3=0\qquad\text{add 3 to both sides}\\\\y=3\\\\y-7=0\qquad\text{add 7 to both sides}\\\\y=7\\\\\text{Put the value of}\ y\ \text{to (1):}\\\\x=10-3=7\ or\ x=10-7=3`