What does it mean to solve a system of equations?

Answer 1

Solving one equation means identifying all the points that make the equation true.

So if the equation is x = y, the solution is the (the set of all points on the) line through the origin at 45 degrees. This is also the same as the set of all points (a,a) for all real values of a.

Solving a set of equations means identifying all the points that make all the equations in the set true.

For example, x^2 + y^2 = 2 is the equation for a circle of radius square root of 2, with center at the origin. There are two points that satisfy the equation of this circle, and the equation of the line above.

They are (1,1) and (-1, -1)

How you solve a system of equations depends on the equations themselves. If the system contains linear equations, then there are standard rules for solving them.

If you have two quadratic equations, such as:
x^2 + y^2 = 25, and
x^2 - y = 5

there are algorithms for finding the "exact" answers, but (almost?) no one uses them.

More generally, there are no exact answers and people use algorithms for approximate (i.e. numerical) answers.

Answer 2

solving a systems of equations is finding the point were the two lines intersect.