Systems of Two Equations with Two Variables. Equivalent Systems

Suppose we are given two equations with two variables `f(x,y)=0` and `g(x,y)=0`. If we need to find all common solutions of two equations with two variables, then it is said that system of equations should be solved. Every pair of values that satisfy all equations in the system, is called a solution of the system of equations.

To solve system means to find all its solutions or prove that there are no solutions.

Two system of equations are called equivalent, if these systems have same solutions. In particular, if both systems don't have solutions, then they are equivalent.

Fact 1. Suppose we are given system of two equations with two variables. If we leave one equation of the system unchanged, and another replace with equivalent equation, then we will obtain equivalent system. For example `{(x+y=1),(x-y=3):}` is equivalent to the `{(x+y=1),(x=3+y):}` because second equations in each system are equivalent

From this fact it follows that if we replace every equation of the system with equivalent equation, then we will obtain equivalent system. For example `{(2x+4y=6),(x-y=3):}` is equivalent to the `{(x=3-2y),(x=3+y):}` because both first and second equations in each system are equivalent.

Fact 2. Suppose we are given system of two equations with two variables. If we leave one equation of the system unchanged, and another replace with sum or difference of both equations in the system, then we will obtain equivalent system. For example `{(x+y=1),(x-y=3):}` is equivalent to the `{(x+y=1),(2x=4):}` because second equation is sum of equations in the system.