Solving Systems of Equations

Definition

• A system of equations is a set of two or more equations containing the same variables
• The solution to a system of equations is a set of values for the variables that satisfies all equations simultaneously
• Systems of equations can be linear or non-linear, depending on the type of equations they contain

Linear Systems of Equations

• A linear system of equations consists of equations where all variables are raised to the first power
• Linear systems can have one solution, no solution, or infinitely many solutions
• The number of equations and variables determines the possible types of solutions

Methods for Solving Linear Systems

• Substitution: Solve one equation for one variable and substitute that expression into the other equation(s)
• Elimination: Add or subtract multiples of the equations to eliminate one variable
• Graphing: Graph each equation and find the point(s) of intersection, which represent the solution(s)
• Matrix Methods: Represent the system as a matrix and use row operations to solve (Gaussian elimination, Gauss-Jordan elimination)
• Cramer's Rule: Use determinants to find the values of the variables

Substitution Method

• Choose one equation and solve for one variable in terms of the other(s)
• Substitute the expression found in the previous step into the other equation(s)
• Solve the resulting equation(s) for the remaining variable(s)
• Substitute the values found back into the expression from the first step to find the value of the other variable

Elimination Method

• Multiply one or both equations by constants so that the coefficients of one variable are opposites
• Add the equations to eliminate the variable with opposite coefficients
• Solve the resulting equation for the remaining variable
• Substitute the value found back into one of the original equations to find the value of the eliminated variable

Graphing Method

• Graph each equation on the same coordinate plane
• Find the point(s) where the graphs intersect
• The coordinates of the intersection point(s) represent the solution(s) to the system
• If the lines are parallel, there is no solution
• If the lines are the same, there are infinitely many solutions

Matrix Methods

• Represent the system of equations as an augmented matrix
• Use row operations to transform the matrix into row-echelon form or reduced row-echelon form (Gaussian elimination or Gauss-Jordan elimination, respectively)
• Solve for the variables using back-substitution

Cramer's Rule

• For a system of n linear equations in n variables, Cramer's Rule can be used to find the solution if the determinant of the coefficient matrix is non-zero
• Calculate the determinant of the coefficient matrix (D)
• Replace the i-th column of the coefficient matrix with the constant terms to form a new matrix, and calculate its determinant (Di)
• The value of the i-th variable is given by xi = Di / D

Types of Solutions

• Unique Solution: The system has exactly one solution (intersecting lines in 2D)
• No Solution: The system has no solutions (parallel lines in 2D)
• Infinitely Many Solutions: The system has an infinite number of solutions (same line in 2D)

Consistent and Inconsistent Systems

• A consistent system has at least one solution (unique or infinitely many)
• An inconsistent system has no solution

Dependent and Independent Systems

• A dependent system has infinitely many solutions
• An independent system has a unique solution

Non-Linear Systems of Equations

• A non-linear system of equations contains at least one non-linear equation (e.g., quadratic, exponential, trigonometric)
• Solving non-linear systems can be more complex than solving linear systems
• Methods for solving include substitution, elimination, and graphing, but may require more advanced techniques
• Non-linear systems can have multiple solutions, no solution, or infinitely many solutions

Applications of Systems of Equations

• Systems of equations are used to model and solve problems in various fields, including physics, engineering, economics, and computer science
• Examples include circuit analysis, mixture problems, supply and demand analysis, and optimization problems