Non-Linear Systems

A non-linear system is a system of equations where at least one equation is not linear, meaning it cannot be represented as a straight line on a graph.

The general form of a quadratic equation is y = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

The vertex of a parabola given by y = ax² + bx + c can be found using the formula x = -b/(2a). Substitute this x-value back into the equation to find the y-coordinate.

A point is a solution to a system of equations if it satisfies all equations in the system, meaning it makes all equations true when substituted.

The discriminant (D = b² - 4ac) determines the nature of the roots of the quadratic equation: D > 0 means two distinct real roots, D = 0 means one real root, and D < 0 means no real roots.

To graph a quadratic function, identify the vertex, axis of symmetry, and intercepts. Plot these points and draw a smooth curve through them.

The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

The method of elimination involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other variable.

Two equations intersect if their graphs meet at one or more points, which corresponds to the solutions of the system of equations.

A parabola is the graph of a quadratic function, characterized by its U-shape, which opens upwards if a > 0 and downwards if a < 0.