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
