Suppose we have system of two equations with two unknowns:
Rewrite the equations in canonical way:
The first equation of the system is an ellipse with a semi-major axis equal to 2 and a semi-minor axis equal to . The second equation is the straight line with tangent equal to and the value of the segment to be cut off on Oy axis equal to .
Let's draw the above on a schematic diagram:
The intersection points of straight line with ellipse M1(x1,y1) and M2(x2,y2) are solutions of the initial system of equations. Since the line intersects the ellipse only in two point indicated above, there are no other solutions.
Just now we have considered so-called graphical method of solving system of equations, which is well suited for solving a system of two equations with two unknowns. As the number of equations increase, the solutions will be points in multidimensional space and it will significantly complicate the task.
If in order to solve the initial system of equations we'll use more universal substitution method, we'll get the following result:
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