Suppose we have straight lines with equation 3y+2x−2=0 and 2y−3x+1=0. Where do the lines intersect?

We have the system of equations

There are two ways of solving this.

By substitution

Rearranging

Substituting this into 3y−3x+1=0:

So x=53​, which implies y=−32​⋅53​+32​=154​. So the intersection is (53​,154​).

By elimination

We can eliminate y from the equations by subtracting the second from the first:

So x=53​⇒y=154​ and the intersection is again (53​,154​).

We can use either of these methods to systems of equations with 2 equations and 2 unknowns.