Given x+y-3=0 is a straight line which intersects the curve x²+y²-xy=21 at two different point. find the coordinates of the points.

Remark
This makes your life really hard when there's an xy term. The easiest way to do this is to graph it. There are some terrific programs on the internet which will do it for you. I use Desmos.  The graph is below. I cannot believe anyone in middle school would be asked to solve a monster like this one using algebra. I'll come back if I think of an easy way to do it. In the meantime, just read the answers where the blue line intersects the red ellipse.

Having said that there is a way that you can solve for xy

Step one
Square x + y=3
x^2 + y^2 + 2xy = 9

Step Two
Subtract the equation  from step one from the equation that is given with xy in it.
x^2 + y^2 - xy = 21
x^2 + y^2 +2xy = 9
-3xy = 12  Divide by -3
xy = 12 / - 3
xy = - 4

Step 2
Use step 2 to find either x or y
x + y = 3
xy = - 4
y = -4/x

x - 4/x = 3   Multiply through by x
x^2 - 4 = 3x  Subtract 3x from both sides.
x^2 - 3x - 4 = 0  Factor this
(x - 4)(x + 1) = 0

x - 4 = 0
x = 4
xy = -4
4*y = - 4 Divide both sides by 4
y = -4/4
y = - 1

Answer 1: One solution is (4,-1) <<<<<< ANSWER

x + 1 = 0
x = - 1
xy = - 4
-1 * y = -4  divide both sides by -1
y = 4

Answer 2: (-1, 4) is  the second answer