The equation of type:
ах² + bxy + c + dx + ey + f = 0
is a general type of quadratic equation with two variables, in which at least one of the coefficients before the quadratic terms a, b and c is nonzero.
is a general type of system of one linear and one quadratic equation with two variables. Such a system is solved by the method of substitution, namely, from the linear equation we express one of the variables and replace it in the square.
Example. Let's solve the system of equations:
2x + y = 1
x² + xy + y² = 1
We express the variable u from the first equation:
у = 1 – 2х
so by substituting u in the second equation with the expression
1 – 2x
х² + x (1 – 2х) + (1 – 2х) ² = 1
after it is solved, Zx – Zx = 0 is obtained, ie 3x (x – 1), whose roots are x = 0, x = = 1.
Substituting these values of x into the equation u = 1 – 2x, we obtain the corresponding values for y, i.e. y₁ = 1, y₂ = -1.
So, the solutions of the system are the ordered pairs of numbers (0, 1) and (1, -1).