Point of intersection of two curves

**celtic1234** · August 7th 2011, 03:30 AM

Hi
I am struggling with this method-my book does not explain it very well.
Can anybody recommend a method to follow? or maybe an example on line?
thanks
John

**Bacterius** · August 7th 2011, 04:32 AM

Let the two curves be defined by the equations $y_1 = f(x_1)$ and $y_2 = g(x_2)$ . The points of intersection of these two curves is defined as the solutions to the conditions $x_1 = x_2$ and $y_1 = y_2$ . Substituting the curve equations into the previous equation, we get $f(x_1) = g(x_2)$ , and since $x_1 = x_2$ , this is equivalent to $f(x_1) = g(x_1)$ . Let $x_1 = x$ for simplicity, and it follows that solving $f(x) = g(x)$ is equivalent to finding the x-coordinates of all the intersections between the two curves.

So for instance if you want to find the points of intersection of the equations:

$y = x^2 + 2x - 1$
$y = 2x + 3$

You will want to solve $x^2 + 2x - 1 = 2x + 3$ , which simplifies to $x^2 - 4 = 0$ . The solutions are therefore $x = \{-2, 2\}$ , and substituting back these x-values into either of the curve equations will give you the y-coordinates of the points of intersection, which are $y = \{-1, 7\}$ respectively. Thus the points of intersection are $(-2, -1)$ and $(2, 7)$ .

**celtic1234** · August 8th 2011, 12:23 PM

thanks for your help
John

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