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?

1 Answer
Dec 2, 2017

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