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Different Systems of Equations

Concept

System of Equations

A system of equations is a set of two or more equations involving the same variables. The solutions to a system are values for these variables that satisfy all the equations simultaneously. The most common systems are systems of linear equations, which contain only linear equations. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side. 2x-3y=1 3x+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. Since the graphs of linear equations are lines, they can have 0, 1, or infinitely many points of intersection. Therefore, the number of solutions for a linear system is also 0, 1, or infinitely many solutions.

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

Extra

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+2y=8 2x-3y=1
Linear-Quadratic System	Includes one linear equation and one quadratic equation.	y=3x+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.	sinx+y=1 x^2+y^2=4

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

Concept

Nonlinear System

A nonlinear system is a system of equations in which at least one of the equations is nonlinear. Consider the following system of equations. 2x + 4y = 10 & (I) 4x^2 + y = - 5 & (II)

Even if the first equation in the system is linear, the system is a nonlinear system because the x-term of Equation (II) is elevated to the power of 2, making it nonlinear.

Concept

System of Quadratic Equations

A system of quadratic equations is a system of equations that consists of only quadratic equations. y=x^2-6x+3 y=- x^2+2x-5 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= - 5 are a solution to the system. This can be verified by substituting the values into each equation.

y=x^2-6x+3	y=- x^2+2x-5
-5 = 2^2-6( 2)+3	-5=- 2^2+2( 2)-5
-5 = 4 - 12 + 3	-5 =-4 + 4-5
-5 = -5	-5 =-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.

Number of solutions for a system of quadratic equations

Quadratic systems can be solved graphically or algebraically. Since the equations in a quadratic system are not linear, these are nonlinear systems.

Concept

Linear-Quadratic System

A linear-quadratic system is a system of equations containing one linear equation and one quadratic equation. 2x-y=3 x^2+y-3x=9 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.

Number of solutions to a linear-quadratic system

A linear-quadratic system can be solved both algebraically and graphically. Since this system includes a quadratic equation, it is a nonlinear system.

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