Determine algebraically all points where the graphs of xy=10 and y=x+3 intersect

2 Intersections
y1=10/x. y2=x+3

10/x=x+3
x=2 clearly => (2,5)
therefore x=-5=>(-5,-2)

You can use substitution and plug in the y value given as x+3 into xy=10.
x(x+3)=10
or
x^2+3x=10
or
x^2+3x-10=0

Then you divide the -10 into two different factors, I did 2 and -5.

x^2 is equal to xx, so now you have one x for each factor of the -10:
(x+2)(x-5)=0

This means x is either 2 or -5, and when you plug in these values to the equation y=x+3, you get:

(2,5)
and
(-5,-2)

Which are your two intersecting points!