Answer :

[tex]x;\ y-the\ numbers\\\\\left\{\begin{array}{ccc}x+y=19&|subtract\ y\ from\ both\ sides\\xy=60\end{array}\right\\\left\{\begin{array}{ccc}x=19-y&(1)\\xy=60&(2)\end{array}\right\\\\subtitute\ (1)\ to\ (2)\\\\(19-y)y=60\\19y-y^2=60\ \ \ \ |subtract\ 60\ from\ both\ sides\\-y^2+19y-60=0\\-y^2+15y+4y-60=0\\-y(y-15)+4(y-15)=0\\(y-15)(-y+4)=0\iff y-15=0\ or\ -y+4=0\\\\\boxed{y=15\ or\ y=4}[/tex]

[tex]subtitute\ the\ value\ of\ "y"\ to\ (1)\\\\x=19-15=4\ or\ x=19-4=15\\\\Answer:\\\boxed{\left\{\begin{array}{ccc}x=4\\y=15\end{array}\right\ or\ \left\{\begin{array}{ccc}x=15\\y=4\end{array}\right\ conclusion:15\ and\ 4}[/tex]
I did this in a different way, not using algebra

I started with the factors of 60
1x60
2x30
3x20
4x15  which are the two numbers that added up to 19

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