Systems of equations involve solving multiple equations simultaneously to find solutions that satisfy all of the equations.
Variables: Understanding what a variable is and how it represents an unknown value in an equation.
Linear Equations: Basic understanding of linear equations and their properties such as slope and intercept.
Systems of Equations: Definition, types, and examples of systems of equations.
Graphing Systems of Equations: Plotting the equations of the system on the same coordinate plane and finding their intersection points.
Substitution Method: Solving a system by expressing one variable in terms of the other and substituting it into the other equation.
Elimination Method: Solving a system by adding or subtracting the equations to eliminate one variable.
Matrices and Determinants: Learning about matrices and how they can be used to solve systems of equations.
Cramer's Rule: A method of solving a system of linear equations using determinants.
Nonlinear Systems of Equations: Introduction to nonlinear equations and how to solve them.
Applications of Systems of Equations: Real-world examples of how systems of equations are used in solving problems such as mixture problems, motion problems, and investment problems.
Inconsistent and Dependent Systems: Understanding the meaning of an inconsistent system or a system that has infinite solutions.
Linear Programming: Introduction to linear programming and how systems of equations are used to optimize constraints and objectives.
Solving Systems of Equations with Technology: Using graphing calculators, online tools, and software to solve systems of equations.
Systems of Equations in Three or More Variables: Explanation and examples of systems of equations with three or more variables.
Advanced Topics: Advanced topics in systems of equations such as eigenvalues, eigenvectors, and eigenspaces, and how they relate to solving systems of equations.
Linear Systems: A system of linear equations is a set of two or more linear equations in the same variables. The goal is to find the values of the variables that satisfy all the equations together.
Nonlinear Systems: A system of nonlinear equations is a set of two or more equations where at least one of the equations is nonlinear. These types of systems may have multiple solutions or no solutions.
Homogeneous Systems: A system of equations is homogeneous if all the constant terms are zero, i.e. the right-hand side of each equation is zero. Homogeneous systems always have at least one solution, which is the trivial solution where all variables are zero.
Inhomogeneous Systems: The opposite of homogeneous systems, inhomogeneous systems, is the one where at least one of the equations has a non-zero constant term. Also, known as non-homogeneous systems, they often have infinitely many solutions.
Consistent Systems: A consistent system is one in which there is at least one solution when the equations are solved together.
Inconsistent Systems: An inconsistent system of equations is one that has no solution. The equations in the system are contradictory.
Uniquely Solvable Systems: A system of equations is uniquely solvable if it has one and only one solution.
Underdetermined Systems: An underdetermined system of equations is one where there are fewer equations than variables. Such systems do not always have unique solutions and may have infinite solutions.
Overdetermined Systems: Overdetermined systems of equations have more equations than variables. Such systems may have no solution or may have more solutions than equations.
Diophantine Systems: Diophantine equations are those where the solutions must be integers or whole numbers. These systems are often used in number theory and cryptography.
Partially Solved Systems: A partially solved system of equations is one in which only some solutions are known, and the goal is to find the rest of the solutions.
Parametric Systems: A parametric system of equations is one where one or more variables is expressed in terms of other variables. The solutions to these systems are often expressed as a set of equations in terms of arbitrary parameters.
Reduced Row Echelon Form Systems: The reduced row echelon form (RREF) of a system of linear equations is a way of solving the equations using matrix methods. Systems in RREF are easier to solve than systems in other forms.