Systems and Sets of Equations

Suppose we are given two equations `f_1(x)=g_1(x)` and `f_2(x)=g_2(x)`. If we need to find values of variable that satisfy both given equations, it is said that system of equations is given: `{(f_1(x)=g_1(x)),(f_2(x)=g_2(x)):}`.

Example 1. Solve system of equations `{(x^2-1=0),((x-1)(x-2)=0):}`.

Roots of first equation are numbers 1 and -1. Roots of second equation are numbers 1 and 2.

Common root is only number 1. Thus, x=1 is solution of given system.

In case we need to find all values of variable that satisfy at least one equation, then we say that set of equations is given.

Example 2. Solve set of equations `x^2-1=0,(x-1)(x-2)=0`.

Roots of first equation are numbers 1 and -1. Roots of second equation are numbers 1 and 2.

Therefore, solutions of the given set are numbers 1,-1 and 2.