Solve the following system of equations
algebraically:

9x² + y² = 9
  3x – y = 3

Let's do it algebraically and then check it
graphically:

Solve the second equation for y

  3x – y = 3
      -y = 3 - 3x

Multiply through by -1

       y = -3 + 3x

Save one sign by swapping terms

       y = 3x - 3
       
Substitute into the 1st equation:

               9x² + y² = 9

        9x² + (3x - 3)² = 9

 9x² + (3x - 3)(3x - 3) = 9

9x² + 9x² - 9x - 9x + 9 = 9

         18x² - 18x + 9 = 9

             18x² - 18x = 0

Factor out 18x

             18x(x - 1) = 0

Set each factor = 0

18x = 0 gives x = 0
x - 1 = 0 gives x = 1

For each of those solutions for x,
we must find a corresponding solution
for y:

To find this we substitute into

         y = 3x - 3 

For x = 0,

         y = 3(0) - 3
         y = -3

So one solution is (x, y) = (0, -3)

For x = 1,

         y = 3(1) - 3
         y = 0

So the other solution is (x, y) = (1, 0)

This means the two graphs would have
two points of intersection. The graph of

9x² + y² = 9

is this oval shaped graph, called an
ellipse:

 

and the graph of 3x – y = 3

is this line which crosses it twice:



Notice that the graphs cross at the points (1, 0)
and (0, -3)

Edwin