## Saturday, October 2, 2010

### Simultaneous Equations

Mary and Johnny are asked to find the points of intersection of the line x + 5y = 17 with the curve x2 + 5y2 = 49.

to see that the curve is an ellipse intersected by the line at two points. Note that Point B is rounded to 2 decimal places.

A zoomed in view:

Mary rearranges the equation of the line to get x = 17 – 5y, which she substitutes into the equation of the ellipse.

x2 + 5y2 = 49
(17 – 5y)2 + 5y2 = 49
289 – 170 y + 25 y2 + 5y2 = 49
30y2 – 170 y + 240 = 0
3y2 – 17 y + 24 = 0
(3y – 8) (y – 3) = 0
y = 8/3 or y = 3

She puts these values of y back into x + 5y = 17
When y = 3, x = 2
When y = 8/3, x = 11/3

Her answer agrees with Geogebra that the points of intersection are A (2, 3) and B (3.67, 2.67).

___________________________________________________________________________
Examples

1. Solve the simultaneous equations:
x + y = 3 (1)
x2 + y2 = 29 (2)

2. Solve the simultaneous equations:
y – x = 3 (1)
xy = 4 (2)

3. Solve the simultaneous equations:
2x + 3y = 14 (1)
xy = 4 (2)

4. The line y = 3x – 1 intersects the curve 11 = 2x2 + 2y2 – x + y at A and B.
Find the co-ordinates of A and B.

5. The line 4x – 3y = 15 intersects the curve 45 = 8x2 – 27y2 at A and B.
Find the co-ordinates of A and B.

6. The line x – 3y = -1 intersects the ellipse 2(x – y) = 2x2 – 11y2 at A and B.
Find the co-ordinates of A and B.