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System of Equations

A system of equations is a set of two or more equations involving the same variables. The solutions to a system of equations are values for these variables that satisfy all the equations simultaneously. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side.

{2 x - 3 y = 1 3 x + y = 7

Graphically, solutions to systems of equations are the points where the graphs of the equations intersect. For this reason, these solutions are usually expressed as coordinates.

Systems can contain many different types of equations. Systems of equations can be solved graphically or algebraically.

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Types of System of Equations

Systems of equations can be classified based on their algebraic characteristics, especially the types of equations involved. The following are some of the most common types.

Type of System	Description	Example
Linear System	Includes only linear equations.	${x + 2 y = 8 2 x - 3 y = 1$
Linear-Quadratic System	Includes one linear equation and one quadratic equation.	${y = 3 x + 1 x^{2} + y^{2} = 25$
Quadratic System	Consists solely of quadratic equations.	${x^{2} + y^{2} = 16 y = x^{2} - 4$
Nonlinear System	Contains at least one nonlinear equation.	${sin x + y = 1 x^{2} + y^{2} = 4$

Keep in mind that linear-quadratic systems and quadratic systems are also nonlinear systems.

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