Question

What happens graphically when you set equations equal to each other?

I know you can solve the equations to find points where they meet, but how exactly does this work graphically?

Answer 1

To solve a system of equations graphically we look for the point of intersection.

Eg. Consider:

`y = x` and `y = 1-x`

graph{(y-x)(y+x-1)=0 [-2.5, 2.5, -1.25, 1.25]}

Using the graphs we would estimate that the point of intersection is `(1/2,1/2)`

The above example is trivial as we can can both confirm the above solution a derive it algebraically.

If however we have more complex function, for example:

`y = lnx` and `y=2-x`

Then we cannot derive an algebraic solution:
graph{(y-lnx)(y+x-2)=0 [-2.5, 2.5, -1.25, 1.25]}

Here we estimate a solution `(1.5,0.5)` and here this estimate will prove very useful as start point an iterative numerical technique for solving equations (eg Newton-Rhapson). It will also provide vital information about the number of solutions