Different Systems of Equations

A nonlinear system is a system of equations in which at least one of the equations is nonlinear. Consider the following system of equations. Even if the first equation in the system is linear, the system is a nonlinear system because the $x\text{-}$term of Equation (II) is elevated to the power of $2,$ making it nonlinear.

A system of quadratic equations is a system of equations that consists of only quadratic equations. Similar to systems of linear equations, the solution to a system of quadratic equations are the values of the variables that make all the equations true. In the example above, $x=2$ and $y=\text{-}5$ are a solution to the system. This can be verified by substituting the values into each equation.

$y=x^2-6x+3$	$y=\text{-}{x}^2+2x-5$
$\text{-}5=2^2-6(2)+3$	$\text{-}5=\text{-}{2}^2+2(2)-5$
$\text{-}5=4-12+3$	$\text{-}5=\text{-}4+4-5$
$\text{-}5=\text{-}5$	$\text{-}5=\text{-}5$

In the examples, since the equations remain true, the values are a solution of the quadratic system. The graphs of quadratic systems with two equations can have anywhere from zero to four points of intersection, or even infinitely many points of intersection. Therefore, quadratic systems may have $0,$ $1,$ $2,$ $3,$ $4,$ or infinitely many solutions. Quadratic systems can be solved graphically or algebraically. Since the equations in a quadratic system are not linear, these are nonlinear systems.

A linear-quadratic system is a system of equations containing one linear equation and one quadratic equation. Similar to a system of linear equations, the solutions to a linear-quadratic system are the values that satisfy both equations simultaneously. For instance, in the given example $x=4$ and $y=5$ are solutions. The values can be verified by substituting them into each equation.

$2x-y=3$	$x^2+y-3x=9$
$2(4)-5=3$	$4^2+5-3(4)=9$
$8-5=3$	$16+5-12=9$
$3=3✓$	$9=9✓$

In the example, since the equations remain true, the values are a solution of the linear-quadratic system. The graph of the linear equation is a straight line, and that of the quadratic equation is a parabola. These graphs can have $0,$ $1,$ or $2$ points of intersection. Therefore, the number of solutions for a linear-quadratic system is also $0,$ $1,$ or $2.$ A linear-quadratic system can be solved both algebraically and graphically. Since this system includes a quadratic equation, it is a nonlinear system.